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Emil de Souza Sánchez Filho
Tensor 
Calculus for 
Engineers and 
Physicists 
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Tensor Calculus for Engineers and Physicists
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Emil de Souza Sánchez Filho
Tensor Calculus
for Engineers and Physicists
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Emil de Souza Sánchez Filho
Fluminense Federal University
Rio de Janeiro, Rio de Janeiro
Brazil
ISBN 978-3-319-31519-5 ISBN 978-3-319-31520-1 (eBook)
DOI 10.1007/978-3-319-31520-1
Library of Congress Control Number: 2016938417
© Springer International Publishing Switzerland 2016
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To
Sandra, Yuri, Nat�alia and Lara
Preface
The Tensor Calculus for Engineers and Physicist provides a rigorous approach to
tensor manifolds and their role in several issues of these professions. With a
thorough, complete, and unified presentation, this book affords insights into several
topics of tensor analysis, which covers all aspects of N-dimensional spaces.
Although no emphasis is placed on special and particular problems of Engineer-
ing or Physics, the text covers the fundamental and complete study of the aim of
these fields of the science. The book makes a brief introduction to the basic concept
of the tensorial formalism so as to allow the reader to make a quick and easy review
of the essential topics that enable having a dominium over the subsequent themes,
without needing to resort to other bibliographical sources on tensors.
This book did not have the framework of a math book, which is a work that
seeks, above all else, to organize ideas and concepts in a didactic manner so as to
allow the familiarity with the tensorial approach and its application of the practical
cases of Physics and the areas of Engineering.
The development of the various chapters does not cling to any particular field of
knowledge, and the concepts and the deductions of the equations are presented so as
to permit engineers and physicists to read the text without being experts in any
branch of science to which a specific topic applies.
The chapters treat the various themes in a sequential manner and the deductions
are performed without omission of the intermediary steps, the subjects being treated
in a didactic manner and supplemented with various examples in the form of solved
exercises with the exception of Chap. 3 that broaches review topics. A few
problems with answers are presented at the end of each chapter, seeking to allow
the reader to improve his practice in solving exercises on the themes that were
broached.
Chapter 1 is a brief introduction to the basic concepts of tensorial formalism so
as to permit the reader to make a quick and easy review of the essential topics that
make possible the knowledge of the subsequent themes that come later, without
needing to resort to other bibliographic sources on tensors.
vii
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The concepts of covariant, absolute, and contravariant derivatives, with the
detailed development of all the expressions concerning these parameters, as well
as the deductions of the Christoffel symbols of the first and second kind, are the
essence of Chap. 2.
Chapter 3 presents the Green, Stokes, and Gauss–Ostrogradsky theorems using a
vectorial formulation.
The expansion of the concepts of the differential operators studied in Differential
Calculus is performed in Chap. 4. The scalar, vectorial, and tensorial fields are
defined, and the concepts and expressions for gradient, divergence, and curl are
formulated. With the definition of the nabla operator, successive applications of this
linear differential operator are carried out and various fundamental relations
between the differential operators are deducted, defining the Laplace operator.
All the formulas are deducted by means of tensorial approach.
The definition of metric spaces with several dimensions, with the introduction of
Riemann curvature concept, and the Ricci tensor formulations, the scalar curvature,
and the Einstein tensor are the subjects studied in detail in Chap. 5. Various
particular cases of Riemann spaces are analyzed, such as the bidimensional spaces,
the spaces with constant curvature, the Minkowski space, and the conformal spaces,
with the definition of the Weyl tensor.
Chapter 6 broaches metric spaces provided with curvature with the introduction
of the concepts of the geodesics and the geodesics and Riemann coordinate
systems. The geodesics deviation and the parallelism of vectors in curved spaces
are studied, with the definition of the torsion tensor concept.
The purpose of this book is to give a simple, correct, and comprehensive
mathematical explanation of Tensor Calculus, and it is self-contained. Postgraduate
and advanced undergraduate students and professionals will find clarity and insight
into the subject of this textbook.
The preparation of a book is a hard and long work that requires the participation
of other people besides the author, which are of fundamental importance in the
preparation of the originals and in the tiresome task of reviewing the typing, chiefly
in a text such as the one in this book. So, our sincere thanks to all those who helped
in the preparation and editing of these pages.
In relation to the errors in this text which were not corrected by a more diligent
review, it is stressed that they are the author’s responsibility and the author
apologizes for them.
Rio de Janeiro, Brazil Emil de Souza Sánchez Filho
December 17, 2015
viii Preface
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Historical Introduction
This brief history of Tensor Calculus broaches the development of the idea of
vector and the advent of the concept of tensor in a synthetic way. The following
paragraphs aim to show the history of the development of these themes in the course
of time, highlighting the main stages that took place in this evolution of the
mathematical knowledge. A few items of bibliographic data of the mathematicians
and scientists who participated on this epic journey in a more striking manner are
described.
The perception of Nature under a purely philosophical focus led Plato in 360 BC
to the study of geometry. This philosopher classified the geometric figures into
triangles, rectangles, and circles, and with this system, he grounded the basic
conceptsof geometry. Later Euclid systemized geometry in axiomatic form,
starting from the fundamental concepts of points and lines.
The wise men of ancient Greece also concerned themselves with the study of the
movement of bodies by means of geometric concepts. The texts of Aristotle (384–
322 BC) inMechanics show that he had the notion of composition of movements. In
this work, Aristotle enounced in an axiomatic form that the force that moves a body
is collinear with the direction of the body’s movement. In a segment ofMechanics,
he describes the velocity of two bodies in linear movement with constant pro-
portions between each other, explaining that “When a body moves with a certain
proportion, the body needs to move in a straight line, and this is the diameter of the
figure formed with the straight lines which have known proportions.” This state-
ment deals with the displacements of two bodies—the Greek sage acknowleding
that the resultant of these displacements would be the diagonal of the rectangle (the
text talks about the diameter) from the composition of the speeds.
In the Renaissance, the prominent figure of Leonardo da Vinci (1452–1519) also
stood out in the field of sciences. In his writings, he reports that “Mechanics is the
paradise of mathematical science, because all the fruits of mathematics are picked
here.” Da Vinci conceived concepts on the composition of forces for maintaining
the balance of the simple structures, but enunciated them in an erroneous and
contradictory manner in view of the present-day knowledge.
ix
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The awakening of a new manner of facing the uniform was already blossoming
in the 1600s. The ideas about the conception and study of the world were no longer
conceived from the scholastic point of view, for reason more than faith had become
the way to new discoveries and interpretations of the outside world. In the Nether-
lands, where liberal ideas were admitted and free thought could be exercised in full,
the Dutch mathematician Simon Stevin (1548–1620), or Stevinus in a Latinized
spelling, was the one who demonstrated in a clear manner the rule for the compo-
sition of forces, when analyzing the balance of a body located in an inclined plane
and supported by weights, one hanging at the end of a lever, and the other hanging
from a pulley attached to the vertical cathetus of the inclined plane. This rule is a
part of the writings of Galileo Galilei (1564–1642) on the balance of bodies in a
tilted plane. However, it became necessary to conceive mathematical formalism
that translated these experimental verifications. The start of the concept of vector
came about in an empirical mode with the formulation of the parallelogram rule, for
Stevinus, in a paper published in 1586 on applied mechanics, set forth this principle
of Classic Mechanics, formalizing by means of the balance of a force system the
concept of a variety depending on the direction and orientation of its action,
enabling in the future the theoretical preparation of the concept of vector.
The creation of the Analytical Geometry by René Du Perron Descartes (1596–
1650) brought together Euclid’s geometry and algebra, establishing a univocal
correspondence between the points of a straight line and the real numbers. The
introduction of the orthogonal coordinates system, also called Cartesian coordi-
nates, allowed the calculation of the distance ds between two points in the Euclid-
ean space by algebraic means, given by ds2 ¼ dx2 þ dy2 þ dz2, where dx, dy, dz are
the coordinates of the point.
The movement of the bodies was a focus of attention of the mathematicians and
scientists, and a more elaborate mathematical approach was necessary when it was
studied. This was taken care of by Leonhard Paul Euler (1707–1783), who con-
ceived the concept of inertia tensor. This concept is present in his book Theoria
Motus Corporum Solidorum seu Rigidorum (Theory of the Movement of the Solid
and Rigid Bodies) published in 1760. In this paper, Euler studies the curvature lines,
initiating the study of Differential Geometry. He was the most published
x Historical Introduction
mathematician of the all time, 860 works are known from him, and it is known that
he published 560 papers during his lifetime, among books, articles, and letters.
In the early 1800s, Germany was becoming the world’s largest center in math-
ematics. Among many of its brilliant minds, it counted Johann Karl Friedrich Gauß
(1777–1855). On occupying himself with the studies of curves and surfaces, Gauß
coined the term non-Euclidean geometries; in 1816, he’d already conceived con-
cepts relative to these geometries. He prepared a theory of surfaces using curvilin-
ear coordinates in the paper Disquisitones Generales circa Superfı́cies Curvas,
published in 1827. Gauß argued that the space geometry has a physical aspect to
be discovered by experimentation. These ideas went against the philosophical
concepts of Immanuel Kant (1724–1804), who preconized that the conception of
the space is a priori Euclidian. Gauß conceived a system of local coordinates system
u, v,w located on a surface, which allowed him to calculate the distance between
two points on this surface, given by the quadratic expression ds2 ¼ Adu2þ
Bdv2 þ Cdw2 þ 2Edu � dvþ 2Fdv � dwþ 2Gdu � dw, where A,B,C,F,G are func-
tions of the coordinates u, v,w.
The idea of force associated with a direction could be better developed analyt-
ically after the creation of the Analytical Geometry by Descartes. The representa-
tion of the complex numbers by means of two orthogonal axes, one axis
representing the real numbers and the other axis representing the imaginary num-
ber, was developed by the Englishman John Wallis (1616–1703). This representa-
tion allowed the Frenchman Jean Robert Argand (1768–1822) to develop in 1778,
in a manner independent from the Dane Gaspar Wessel (1745–1818), the mathe-
matical operations between the complex numbers. These operations served as a
framework for the Irish mathematician William Rowan Hamilton (1805–1865) to
develop a more encompassing study in three dimensions, in which the complex
numbers are contained in a new variety: the Quaternions.
Historical Introduction xi
This development came about by means of the works of Hamilton, who had the
beginning of his career marked by the discovery of an error in the bookMécanique
Celeste authored by Pierre Simon-Laplace (1749–1827), which gave him prestige
in the intellectual environment. In his time, there was a great discrepancy between
the mathematical production from the European continent and from Great Britain,
for the golden times of Isaac Newton (1642–1727) had already passed. Hamilton
studied the last advances of the continental mathematics, and between 1834 and
1835, he published the books General Methods in Dynamics. In 1843, he published
the Quaternions Theory, printed in two volumes, the first one in 1853 and the
second one in 1866, in which a theory similar to the vector theory was outlined,
stressing, however, that these two theories differ in their grounds.
In the first half of the nineteenth century, the German Hermann G€unther
Graßmann (1809–1877), a secondary school teacher of the city of Stettin located
in the region that belongs to Pomerania and that is currently a part of Poland,
published the book Die Lineale Ausdehnunsgleher ein neuer Zweig der Mathematik
(Extension Theory), in which he studies a geometry of more than three dimensions,
treating N dimensions, and formulating a generalization of the classic geometry. To
outline this theory, he used the concepts of invariants (vectors and tensors), which
later enabled other scholars to develop calculus and vector analysis.
xii Historical Introduction
The great mathematical contribution of the nineteenth century, which definitely
marked the development of Physics, is due to Georg Friedrich Bernhard Riemann
(1826–1866). Riemann studiedin G€ottingen, where he was a pupil of Gauß, and
afterward in Berlin, where he was a pupil of Peter Gustav Lejeune Dirichlet (1805–
1859), and showed an exceptional capacity for mathematics when he was still
young. His most striking contribution was when he submitted in December 1853
his Habilititationsschrift (thesis) to compete for the position of Privatdozent at the
University of G€ottingen. This thesis titled €Uber die Hypothesen welche der
Geometrie zu Grunde liegen enabled a genial revolution in the structure of Physics
in the beginning of the twentieth century, providing Albert Einstein (1879–1955)
with the mathematical background necessary for formulating his Theory of Rela-
tivity. The exhibition of this work in a defense of thesis carried out in June 10, 1854,
sought to show his capacity to teach. Gauß was a member of examination board and
praised the exhibition of Riemann’s new concepts. His excitement for the new
formulations was expressed in words: “. . . the depth of the ideas that were
presented. . ..” This work was published 14 years later, in 1868, two years after
the death of its author. Riemann generalized the geometric concepts of Gauß,
conceiving a system of more general coordinates spelled as dxi, and established a
fundamental relation for the space of N dimensions, where the distance between
two points ds is given by the quadratic form ds2 ¼ gijdxidxj, having gij a symmet-
rical function, positive and defined, which characterizes the space in a unique
manner. Riemann developed a non-Euclidean, elliptical geometry, different from
the geometries of János Bolyai and Nikolai Ivanovich Lobachevsky. The Riemann
Geometry unified these three types of geometry and generalized the concepts of
curves and surfaces for hyperspaces.
The broaching of the Euclidean space in terms of generic coordinates was
carried out for the first time by Gabriel Lamé (1795–1870) in his work Leçons
sur les Fonctions Inverses des Transcedentes et les Surfaces Isothermes, published
in Paris in 1857, and in another work Leçons sur les Coordonées Curvilignes,
published in Paris in 1859.
Historical Introduction xiii
The new experimental discoveries in the fields of electricity and magnetism
made the development of an adequate mathematical language necessary to translate
them in an effective way. These practical needs led the North American Josiah
Willard Gibbs (1839–1903) and the Englishman Olivier Heaviside (1850–1925), in
an independent manner, to reformulate the conceptions of Graßmann and Hamilton,
creating the vector calculus. Heaviside had thoughts turned toward the practical
cases and sought applications for the vectors and used vector calculus in electro-
magnetism problems in the industrial areas.
With these practical applications, the vectorial formalism became a tool to be
used in problems of engineering and physics, and Edwin Bidwell Wilson, a pupil of
Gibbs, developed his master’s idea in the book Vector Analysis: A Text Book for the
Use for Students of Mathematics and Physics Founded upon Lectures of Josiah
Willard Gibbs, published in 1901 where he disclosed this mathematical apparatus,
making it popular. This was the first book to present the modern system of vectorial
analysis and became a landmark in broadcasting the concepts of calculus and
vectorial analysis.
xiv Historical Introduction
The German mathematician and prominent professor Elwin Bruno Christoffel
(1829–1900) developed researches on the Invariant Theory, writing six articles
about this subject. In the article €Uber die Transformation der Homogenel
Differentialausdr€ucke zweiten Grade, published in the Journal f€ur Mathematik,
70, 1869, he studied the differentiation of the symmetric tensor gij and introduced
two functions formed by combinations of partial derivatives of this tensor, con-
ceiving two differential operators called Christoffel symbols of the first and second
kind, which are fundamental in Tensorial Analysis. With this, he contributed in a
fundamental way to the arrival of Tensor Calculus later developed by Gregorio
Ricci-Curbastro and Tullio Levi Civita. The metrics of the Riemann spaces and the
Christoffel symbols are the fundaments of Tensor Calculus.
The importance of tensors in problems of Physics is due to the fact that physical
phenomena are analyzed by means of models which include these varieties, which
are described in terms of reference systems. However, the coordinates which are
described in terms of the reference systems are not a part of the phenomena, only a
tool used to represent them mathematically. As no privileged reference systems
exist, it becomes necessary to establish relations which transform the coordinates
from one referential system to another, so as to relate the tensors’ components.
These components in a coordinate system are linear and homogeneous functions of
the components in another reference system.
The technological development at the end of the nineteenth century and the great
advances in the theory of electromagnetism and in theoretical physics made the
conception of a new mathematical tool which enabled expressing new concepts and
Historical Introduction xv
laws imperious. The vectorial formalism did not fulfill the broad field and the
variety of new knowledge that needed to be studied more and interpreted better.
This tool began to be created by the Italian mathematician Gregorio Ricci-
Curbastro (1853–1925), who initiated the conception of Absolute Differential
Calculus in 1884. Ricci-Curbastro was a mathematical physicist par excellence.
He was a pupil of the imminent Italian professors Enrico Betti (1823–1892) and
Eugenio Beltrami (1835–1900). He occupied himself mainly with the Riemann
geometry and the study of the quadratic differential form and was influenced by
Christoffel’s idea of covariant differentiation which allowed achieving great
advances in geometry. He created a research group in which Tullio Levi-Civita
participated and worked for 10 years (1887–1896) in the exploration of the new
concepts and of an elegant and synthetic notation easily applicable to a variety of
problems of mathematical analysis, geometry, and physics. In his article,Méthodes
de Calcul Differéntiel Absolu et leurs Applications, published in 1900 in vol. 54 of
the Mathematische Annalen, in conjunction with his pupil Levi-Civita, the appli-
cations of the differential invariants were broached, subject of the Elasticity
Theory, of the Classic Mechanics and the Differential Geometry. This article is
considered as the beginning of the creation of Tensor Calculus. He published the
first explanation of his method in the Volume XVI of the Bulletin des Sciences
Mathématiques (1892), applying it to problems from Differential Geometry to
Mathematical Physics. The transformation law of a function system is due to
Ricci-Curbastro, who published it in an article in 1887, and which is also present
in another article published 1889, in which he introduces the use of upper and lower
indexes, showing the differences between the contravariant and covariant transfor-
mation laws. In these papers, he exhibits the framework of Tensor Calculus.
The pupil and collaborator of Ricci-Curbastro, Tullio Levi-Civita (1873–1941)
published in 1917 in the Rediconti del Circolo Matem�atico di Palermo, XLII
(pp. 173–215) the article Nozione di Parallelismo in una Variet�a Qualunque e
Conseguente Specificazione Geometrica della Curvatura Riemanniana, contribut-
ing in a considerable way to the development of Tensor Calculus. In this work, he
describes the parallelism in curved spaces. This study was presented in lectures
addressed in two courses given at the University of Rome in the period of
xvi Historical Introduction
1920–1921 and 1922–1923. He corresponded with Einstein, who showed great
interest in the new mathematical tool. In 1925, he published the book Lezione di
Calcolo Differenziale Absoluto which is a classic in the mathematicalliterature.
It was the German Albert Einstein in 1916 who called the Absolute Differential
Calculus of Ricci-Curbastro and Levi-Civita Tensor Calculus, but the term tensor,
such as it is understood today, had been introduced in the literature in 1908 by the
physicist and crystallographer G€ottingen, Waldemar Voigt (1850–1919). The
development of the theoretical works of Einstein was only possible after he became
aware of by means of his colleague from Zurich, Marcel Grossmann (1878–1936),
head professor of descriptive geometry at the Eidgen€ossische Technische
Hochschule, the article Méthodes de Calcul Differéntiel Absolut, which provided
him the mathematical tool necessary to conceive his theory, publishing in 1916 in
the Annalen der Physik the article Die Grundlagen der algemeinnen
Relativitatstheorie. His contribution Tensor Calculus also came about with the
conception of the summation rule incorporated to the index notation. The term
tensor became popular mainly due to the Theory of Relativity, in which Einstein
used this denomination. His researches on the gravitational field also had the help of
Grossmann, Tulio Levi-Civita, and Gregorio Ricci-Curbastro, conceiving the Gen-
eral Relativity Theory. On the use of the Tensor Calculus in his Gravitation Theory,
Einstein wrote: “Sie bedeutet einen wahren Triumph der durch Gauss, Riemann,
Christoffel, Ricci . . . begr€undeten Methoden des allgemeinen Differentialkalculus.”
Historical Introduction xvii
Other notable mathematics contributed to the development of the study of
tensors. The Dutch Jan Arnoldus Schouten (1873–1941), professor of the T. U.
Delft, discovered independently of Levi-Civita the parallelism and systematized the
Tensor Calculus. Schouten published in 1924 the book Ricci-Kalk€ulwhich became a
reference work on the subject, where he innovates the tensorial notation, placing the
tensor indexes in brackets to indicate that it was an antisymmetric tensor.
The Englishman Arthur Stanley Eddington (1882–1944) conceived new in
Tensor Calculus and was major promoter of the Theory of Relativity to the lay
public.
The German Hermann Klaus Hugo Weyl (1885–1955) published in 1913 Die
Idee der Riemannschen Fl€ache, which gave a unified treatment of Riemann
xviii Historical Introduction
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surfaces. He contributed to the development and disclosure of Tensor Calculus,
publishing in 1918 the book Raum-Zeit-Materie a classic on the Theory of Rela-
tivity. Weyl was one of the greatest and most influential mathematicians of the
twentieth century, with broad dominium of themes with knowledge nearing the
“universalism.”
The American Luther Pfahler Eisenhart (1876–1965) who contributed greatly to
semi-Riemannian geometry wrote several fundamental books with tensorial
approach.
The work of French mathematician Élie Joseph Cartan (1869–1951) in differ-
ential forms, one of the basic kinds of tensors used in mathematics, is principal
reference in this theme. He published the famous book Leçons sur la Géométrie des
Espaces de Riemann (first edition in 1928 and second edition in 1946).
Historical Introduction xix
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Contents
1 Review of Fundamental Topics About Tensors . . . . . . . . . . . . . . . . . 1
1.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Index Notation and Transformation of Coordinates . . . . . 1
1.2 Space of N Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.2 Kronecker Delta and Permutation Symbol . . . . . . . . . . . . 3
1.3.3 Dual (or Reciprocal) Basis . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.4 Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Homogeneous Spaces and Isotropic Spaces . . . . . . . . . . . . . . . . . 16
1.5 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 Conjugated Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.2 Dot Product in Metric Spaces . . . . . . . . . . . . . . . . . . . . . 30
1.6 Angle Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.1 Symmetrical and Antisymmetrical Tensors . . . . . . . . . . . 43
1.7 Relative Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.7.1 Multiplication by a Scalar . . . . . . . . . . . . . . . . . . . . . . . 54
1.8 Physical Components of a Tensor . . . . . . . . . . . . . . . . . . . . . . . . 62
1.8.1 Physical Components of a Vector . . . . . . . . . . . . . . . . . . 62
1.9 Tests of the Tensorial Characteristics of a Variety . . . . . . . . . . . . 66
2 Covariant, Absolute, and Contravariant Derivatives . . . . . . . . . . . . 73
2.1 Initial Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 Cartesian Tensor Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2.2 Cartesian Tensor of the Second Order . . . . . . . . . . . . . . . 77
2.3 Derivatives of the Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.1 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3.2 Relation Between the Christoffel Symbols . . . . . . . . . . . 83
2.3.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3.4 Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . 84
xxi
2.3.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.3.6 Number of Different Terms . . . . . . . . . . . . . . . . . . . . . . 85
2.3.7 Transformation of the Christoffel Symbol of First Kind . . 86
2.3.8 Transformation of the Christoffel Symbol of Second Kind 87
2.3.9 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.3.10 Orthogonal Coordinate Systems . . . . . . . . . . . . . . . . . . . 88
2.3.11 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.3.12 Christoffel Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.3.13 Ricci Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.3.14 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.4 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.4.1 Contravariant Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.4.2 Contravariant Tensor of the Second-Order . . . . . . . . . . . 104
2.4.3 Covariant Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.4.4 Mixed Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.4.5 Covariant Derivative of the Addition, Subtraction, and
Product of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.4.6 Covariant Derivative of Tensors gij, g
ij, δij . . . . . . . . . . . . 117
2.4.7 Particularities of the Covariant Derivative . . . . . . . . . . . . 121
2.5 Covariant Derivative of Relative Tensors . . . . . . . . . . . . . . . . . . . 123
2.5.1 Covariant Derivative of the Ricci Pseudotensor . . . . . . . . 125
2.6 Intrinsic or Absolute Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.6.1 Uniqueness of the Absolute Derivative . . . . . . . . . . . . . . 131
2.7 Contravariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.1.1 Smooth Surface . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.1.2 Simply Connected Domain . . . . . . . . . . . . . . . . . . . . . . . 137
3.1.3 Multiply Connected Domain . . . . . . . . . . . . . . . . . . . . . 138
3.1.4 Oriented Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.1.5 Surface Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.1.6 Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.2 Oriented Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.2.1 Volume Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.4 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.5 Gauß–Ostrogradsky Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.1 Scalar, Vectorial, and Tensorial Fields . . . . . . . . . . . . . . . . . . . . . 155
4.1.1 Initial Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.1.2 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
xxii Contents
4.1.3 Pseudoscalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.1.4 Vectorial Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.1.5 Tensorial Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.1.6 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.2.1 Norm of the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.2.2 Orthogonal Coordinate Systems . . . . . . . . . . . . . . . . . . . 165
4.2.3 Directional Derivative of the Gradient . . . . . . . . . . . . . . 166
4.2.4 Dyadic Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.2.5 Gradient of a Second-Order Tensor . . . . . . . . . . . . . . . . 169
4.2.6 Gradient Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.3 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.3.1 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.3.2 Contravariant and Covariant Components . . . . . . . . . . . . 179
4.3.3 Orthogonal Coordinate Systems . . . . . . . . . . . . . . . . . . . 181
4.3.4 Physical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.3.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.3.6 Divergence of a Second-Order Tensor . . . . . . . . . . . . . . 183
4.4 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.4.1 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.4.2 Orthogonal Curvilinear Coordinate Systems . . . . . . . . . . 201
4.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.4.4 Curl of a Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.5 Successive Applications of the Nabla Operator . . . . . . . . . . . . . . 207
4.5.1 Basic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4.5.2 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
4.5.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.5.4 Orthogonal Coordinate Systems . . . . . . . . . . . . . . . . . . . 218
4.5.5 Laplacian of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.5.6 Curl of the Laplacian of a Vector . . . . . . . . . . . . . . . . . . 219
4.5.7 Laplacian of a Second-Order Tensor . . . . . . . . . . . . . . . . 220
4.6 Other Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.6.1 Hesse Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.6.2 D’Alembert Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5 Riemann Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2 The Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.2.2 Differentiation Commutativity . . . . . . . . . . . . . . . . . . . . 231
5.2.3 Antisymmetry of Tensor Ri‘jk . . . . . . . . . . . . . . . . . . . . . 233
5.2.4 Notations for Tensor Ri‘jk . . . . . . . . . . . . . . . . . . . . . . . . 233
5.2.5 Uniqueness of Tensor R‘ijk . . . . . . . . . . . . . . . . . . . . . . . 234
5.2.6 First Bianchi Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Contents xxiii
5.2.7 Second Bianchi Identity . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.2.8 Curvature Tensor of Variance (0, 4) . . . . . . . . . . . . . . . . 238
5.2.9 Properties of Tensor Rpijk . . . . . . . . . . . . . . . . . . . . . . . . 240
5.2.10 Distinct Algebraic Components of Tensor Rpijk . . . . . . . . 241
5.2.11 Classification of Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.3 Riemann Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3.2 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.3.3 Normalized Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5.4 Ricci Tensor and Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . 250
5.4.1 Ricci Tensor with Variance (0, 2) . . . . . . . . . . . . . . . . . . 251
5.4.2 Divergence of the Ricci Tensor with Variance
Ricci (0, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.4.3 Bianchi Identity for the Ricci Tensor
with Variance (0, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.4.4 Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.4.5 Geometric Interpretation of the Ricci Tensor
with Variance (0, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.4.6 Eigenvectors of the Ricci Tensor with Variance (0, 2) . . . 256
5.4.7 Ricci Tensor with Variance (1, 1) . . . . . . . . . . . . . . . . . . 257
5.4.8 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
5.5 Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.6 Particular Cases of Riemann Spaces . . . . . . . . . . . . . . . . . . . . . . 264
5.6.1 Riemann Space E2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
5.6.2 Gauß Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
5.6.3 Component R1212 in Orthogonal Coordinate Systems . . . . 269
5.6.4 Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5.6.5 Riemann Space with Constant Curvature . . . . . . . . . . . . 273
5.6.6 Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
5.6.7 Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
5.6.8 Conformal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
5.7 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6 Geodesics and Parallelism of Vectors . . . . . . . . . . . . . . . . . . . . . . . . 295
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.2.1 Representation byMeans of Curves in the Surfaces . . . . . 299
6.2.2 Constant Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
6.2.3 Representation by Means of the Unit Tangent Vector . . . 301
6.2.4 Representation by Means of an Arbitrary Parameter . . . . 302
6.3 Geodesics with Null Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
6.4 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
6.4.1 Geodesic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 309
6.4.2 Riemann Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
6.5 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
xxiv Contents
6.6 Parallelism of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6.6.1 Initial Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6.6.2 Parallel Transport of Vectors . . . . . . . . . . . . . . . . . . . . . 321
6.6.3 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Contents xxv
Notations
ℜ Set of the real numbers
Z Set of the complex numbers
� � �j j Determinant
� � �k k Modulus, absolute value
· Dot product, scalar product, inner product
� Cross product, vectorial product
� Tensorial product
� Two contractions of the tensorial product
δij, δ
ij, δji Kronecker delta
δij . . .m, δ
ij . . .m,
δj:::: ni ...m
Generalized Kronecker delta
eijk, e
ijk Permutation symbol
eijk . . .m, e
ijk . . .m Generalized permutation symbol
εijk, ε
ijk Ricci pseudotensor
εi1i2i3���in , ε
i1i2i3���in Ricci pseudotensor for the space EN
E3 Euclidian space
J Jacobian
EN Vectorial space or tensorial space with N dimension
‘n� � � Natural logarithm
εijk . . .m, ε
ijk . . .m Ricci pseudotensor for the space EN
d . . .
dxk
Differentiation with respect to variable xk
ϕ, i Comma notation for differentiation
_x: Differentiation with respect to time
∂ . . .
∂xk
Partial differentiation with respect to variable xk
∂k� � � Covariant derivative
δ� � �
δ t
Intrinsic or absolute derivative
∇� � � Nabla operator
xxvii
∇2� � � Laplace operator, Laplacian
H� � � Hesse operator, Hessian
□. . . D’Alembert operator, D’alembertian
div� � � Divergent
grad� � � Gradient
lap� � � Laplacian
rot� � � Rotational, curl
gij, g
ij, gji Metric tensor
Γij,k Christoffel symbol of first kind
Γmip Christoffel symbol of the second kind
Gij, G
ij Einstein tensor
Gkm Einstein tensor with variance (1,1)
K Riemann curvature
R Scalar curvature
Rij Ricci tensor of the variance (0,2)
Rij Ricci tensor of the variance (1,1)
R‘ijk Riemann–Christoffel curvature tensor, Riemann–Christoffel
mixed tensor, Riemann–Christoffel tensor of the second kind,
curvature tensor
Rpijk Curvature tensor of variance (0, 4)
tr� � � Trace of the matrix
Wijk‘ Weyl curvature tensor
Greek Alphabets
Sound Letter
Alpha α, Α
Beta β, Β
Gamma γ, Γ
Delta δ, Δ
Epsilon ε, Ε
Zeta ζ, Ζ
Eta η, Η
Theta θ, Θ
Iota ι, I
Kappa κ, K
Lambda λ, Λ
M€u μ, M
N€u ν, Ν
Ksi ξ, Ξ
Omicron o, Ο
Pi π, Π
Rho ρ, Ρ
xxviii Notations
Sigma σ, Σ
Tau τ, Τ
Üpsı́lon υ, Υ
Phi φ,ϕ, Φ
Khi χ, Χ
Psi ψ , Ψ
Omega ω, Ω
Notations xxix
Chapter 1
Review of Fundamental Topics About
Tensors
1.1 Preview
This chapter presents a brief review of the fundamental concepts required for the
consistent development of the later chapters. Various subjects are admitted as being
previously known, which allows avoiding demonstrations that overload the text. It
is assumed that the reader has full knowledge of Differential and Integral Calculus,
Vectorial Calculus, Linear Algebra, and the fundamental concepts about tensors
and dominium of the tensorial formalism. However, are presented succinctly the
essential topics for understanding the themes that are developed in this book.
1.1.1 Index Notation and Transformation of Coordinates
On the course of the text, when dealing with the tensorial formulations, the index
notation will be preferably used, and with the summation rule, for instance,
yj ¼
X3
i¼1
X3
j¼1
aijxi ¼ aijxi, where i is a free index and j is a dummy in the sense
that the sum is independent of the letter used, this expression takes the forms
y1 ¼ a11x1 þ a12x2 þ a13x3
y2 ¼ a21x1 þ a22x2 þ a23x3
y3 ¼ a31x1 þ a32x2 þ a33x3
8><>: )
y1
y2
y3
8><>:
9>=>; ¼
a11 a12 a13
a21 a22 a23
a31 a32 a33
264
375 x1x2
x3
8><>:
9>=>;
The transformation of the coordinates from a point in the coordinate system Xi to
the coordinate system X
i
given by xi ¼ aijxj þ ai0 where the terms aij, ai0 are
constants is called affine transformation (linear). In this kind of transformation,
the points of the space E3 are transformed into points, the straight lines in straight
© Springer International Publishing Switzerland 2016
E. de Souza Sánchez Filho, Tensor Calculus for Engineers and Physicists,
DOI 10.1007/978-3-319-31520-1_1
1
lines, and the planes in planes. When ai0 ¼ 0, this transformation is called linear
and homogeneous. The term ai0 represents only a translation of the origin of the
referential.
1.2 Space of N Dimensions
The generalization of the Euclidian space at three dimensions for a number N of
dimensions is prompt, defining a space EN. This expansion of concepts requires
establishing a group of N variables xi, i ¼ 1, 2, 3, . . .N, relative to a point
P xið Þ2EN , related to a coordinate system Xi, which are called coordinates of the
point in this reference system. The set of points associated in a biunivocal way to
the coordinates of the reference system Xi defines the N-dimensional space EN.
In an analogous way a subspace EM � EN is defined, with M < N, in which the
group of pointsP xið Þ2EM is related biunivocally with the coordinates defined in the
coordinate system Xi. To make a few specific studies easier, at times the space is
divided into subspaces. The space EN is called affine space, and if it is linked to the
notion of distance between two points, then it is a metric space.
1.3 Tensors
1.3.1 Vectors
The structure of a vectorial space is defined by two algebraic operations: (a) the sum
of the vectors and (b) the multiplication of vector by scalar.
The conception of vectors u, v,w as geometric varieties is extended to a broad
range of functions, as long as the set of these functions forms a vectorial space
(linear space) on a set of scalars (numbers). The functions f, g, h, . . .with continuous
derivatives that fulfill certain axioms are assumed as vectors, and all the formula-
tions and concepts developed for the geometric vectors apply to these formulations.
A vectorial space is defined by the following axioms:
1. uþ v ¼ vþ u or f þ g ¼ gþ f .
2. uþ vð Þ þ w ¼ uþ vþ wð Þ or f þ gð Þ þ h ¼ f þ gþ hð Þ.
3. The null vector is such that uþ 0 ¼ u or 0þ f ¼ f .
4. To every vector u there is a corresponding unique vector �u, such that
�uð Þ þ u ¼ 0 or �fð Þ þ f ¼ 0.
5. 1 � uk k ¼ u or 1 � fk k ¼ fk k.
6. m nuð Þ ¼ mn uð Þ or m nfð Þ ¼ mn fð Þ, where m, n are scalars.
7. mþ nð Þu ¼ muþ nu or mþ nð Þf ¼ mf þ nf .
8. m uþ vð Þ ¼ muþ mv or m f þ gð Þ ¼ mf þ mg.
2 1 Review of Fundamental Topics About Tensors
1.3.2 Kronecker Delta and Permutation Symbol
The Kronecker delta is defined by
δ ¼ δij ¼ δ ij ¼
1, i ¼ j
0, i 6¼ j
(
ð1:3:1Þ
that is symmetrical, i.e., δij ¼ δji,8i, j. The Kronecker delta is the identity tensor.
This tensor is used as a linear operator in algebraic developments, such as
∂xi
∂xj
δki ¼ ∂x
k
∂xj
∂xj
∂xi
δki ¼ ∂x
j
∂xk
Tijδ
ikuk ¼ Tkjuk ¼ Tj
The permutation symbol is defined by
eijk ¼ eijk ¼
1 is an even permutation of the indexes
�1 is an odd permutation of the indexes
0 when there are repeated indexes
8><>: ð1:3:2Þ
and the generalized permutation symbol is given by
ei1i2i3���in ¼ ei1i2i3���in¼
1 is an even permutation of the indexes
�1 is an odd permutation of the indexes
0 when there are repeated indexes
8><>: ð1:3:3Þ
Figure 1.1 shows an illustration how to obtain the values of this symbol.
1.3.3 Dual (or Reciprocal) Basis
The vector u expresses itself in the Euclidean space E3 by means of the linear
combination of three linearly independent unit vectors, which form the basis of this
space. For the case of oblique coordinate systems, there are two kinds of basis
2 2
1 1
-1+1
3 3
Fig. 1.1 Values of the
permutation symbol
1.3 Tensors 3
called reciprocal or dual basis. Let vector u expressed by means of their compo-
nents relative to a coordinate system with orthonormal covariant basis ej:
u ¼ ujej
and with ei � ek ¼ δjk, the dot product takes the form u � ek ¼ ujej � ek ¼ ujδjk ¼ uk,
which are the components’ covariant of the vector u. These components are the
projections of this vector on the coordinate axes.
In the case of oblique coordinate system, the basis ej, ek is called reciprocal basis,
which fulfills the condition ej � ek ¼ δ kj . In Fig. 1.2 the axes OXi and OXk are
perpendicular, as are also the axes OXk and OXi.
This definition shows that the dot product of two reciprocal basis fulfills
eik k ei
�� �� cos 90o � αð Þ ¼ 1 > 0 ) ei�� �� ¼ 1
eik k sin α
and with eik k ¼ 1 results in ei
�� �� > 1, then ei and ek have different scales. Let the
representation of the vector u in a coordinate system with covariant basis ei, ej, ek,
where the indexes of the vectors of the basis indicate a cyclic permutation of i, j, k;
thus, u ¼ uiei. These vectors do not have to be coplanar ei � ek � ek 6¼ 0; thus, the
volume of the parallelepiped is given by the mixed product ei � ek � ek ¼ V and
with the relation between the two reciprocal basis ei � ej ¼ δij follows
iX
i
X
k
X
k
X
O
ie
k
e
ie
ke
Fig. 1.2 Reciprocal basis
4 1 Review of Fundamental Topics About Tensors
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1
ei
¼ e i ¼ ej � ek
V
Then vector u in terms of reciprocal basis is defined by u ¼ ujej where uj is the
components of this vector in the new basis (contravariant), having these new
components expressed in terms of the original components.
Consider the representation u in terms of the two basis u ¼ uiei ¼ ujej and with
the dot product of both sides of this expression by ej, and applying the definition of
reciprocal basis e j � ei ¼ 1 provides
uj ¼ uiej � ei
In an analogous way
V ¼ e i � ej � ek
whereV is the volume of the parallelepiped defined by the mixed product of the unit
vectors of the reciprocal basis. The height of the parallelepiped defined by the
mixed product of the unit vectors of a base is collinear with one of the unit vectors
of the reciprocal basis (Fig. 1.3).
The volume of the parallelepiped is determined by means of the mixed product
of three vectors and allows assessing the relations between the same by means of
the reciprocal basis in the levorotatory and dextrorotatory coordinates systems.
Consider the mixed product of the vectors of the basis of a levorotatory coordi-
nate system
V ¼ ei � ej � ek ¼ ei � e123 e2e3eið Þ
which will cancel itself only if i ¼ 1, whereby
V ¼ e123 e1ð Þ2e2e3 ¼ e1ð Þ2e2e3 ) e1ð Þ2 ¼ V
e2e3
h
1
e
2
e
3e
1e
2e
3e
Fig. 1.3 Parallelepiped
defined by means of the
reciprocal basis
1.3 Tensors 5
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and for the reciprocal basis
e1
� � 2 ¼ V
e1ð Þ 2
¼ e2e3
V
V ¼ ei � ej � ek ¼ ei � e123e2e3ei
� � ¼ e1� � 2e2e3
e1 ¼ 1
e1
V ¼ 1
e1
� �
e2e3 ¼ e2e3
V
� �
e2e3 ¼ 1
V
) VV ¼ 1e1 ¼ 1
e1
For a dextrorotatory coordinate system
V ¼ e2 � ej � e3 ¼ e2 � e123 eie3e2ð Þe1 ¼ 1
e1
which cancels itself for i ¼ 1, so
V ¼ e2 � �eie3e2ð Þ ¼ e2ð Þ2e1e3e1 ¼ 1
e1
1
e2ð Þ2
¼ � e1e3
V
e1 ¼ 1
e1
and for the case of reciprocal basis
V ¼ e2 � ei � e3 ¼ e2 e123 eie3e2
� � ¼ � e2� �2e1e3
e2 ¼ � 1
e2
In an analogous way
VV ¼ 1
If e1, e3, e2 are the unit vectors of an orthogonal coordinate system, then the
reciprocal basis e1, e2, e3 also defines this coordinate system.
1.3.3.1 Orthonormal Basis
If the basis is orthonormal
ei � ej � ek ¼ V ¼ V ¼ 1 ei ¼ ej � ek ¼ ei uj ¼ ui
6 1 Review of Fundamental Topics About Tensors
This shows that for the Cartesian vectors, it is indifferent, covariant, or
contravariant, of which the basis is adopted. The vector components in terms of
this basis are equal, and the orthonormal basis is defined by their unit vectors
ei ¼ ui
uik k
The linear transformations 8m, u, v2E3: (a) F muð Þ ¼ mF uð Þ; (b)F u � vð Þ ¼ F mð Þ
defined in the Euclidean space E3 are also defined in the vectorial space E
�
3, for there
is an intrinsic correspondence between these two spaces. The rules of calculus in E�3
are analogous to those of E3, so these parameters are isomorphous.
The existence of this duality is extended to the case of a vectorial space of finite
dimension E�N , having E
�
N �ℜ or E*N � Z, for this space is dual to the Euclidean
space EN.
1.3.3.2 Transformation Law of Vectors
The transformation of the coordinates from one point in the coordinate system Xi to
the coordinate system X
i
is given by xi ¼ ∂xi∂xj xj, where ∂x
i
∂xj
¼ cos αij are the matrix
rotation elements, and its terms are the director cosines of the angles between the
coordinate axes.
In this linear and homogeneous transformation, the points of the space E3 are
transformed into points expressed in terms of the new coordinates. Thus, the unit
vectors ofX
i
and ofX
i
transform according to the law ei ¼ ∂xj∂xi ej, where the values of
∂xi
∂xj
¼ cos xixj are the components of the unit vectors ēi in the coordinate system Xi.
For the position vector, u provides ui ¼ ∂xj∂xi uj. In the case of the inverse transfor-
mation, i.e., of X
i
to Xi, provides analogously ej ¼ ∂xi∂xj ei, following for the
components ∂x
j
∂xi ¼ cos xjxi of the unit vectors ej in the coordinate system X
i
.
The determinant of the rotation matrix ∂x
i
∂xj
			 			 assumes the value þ1 in the case of
the transformation taking place between coordinate systems of the same direction,
which is then called proper transformation (rotation). Otherwise ∂x
i
∂xj
			 			 ¼ �1, and the
transformation is called improper transformation (reflection).
1.3.3.3 Covariant and Contravariant Vectors
The representation of the vectors in oblique coordinate systems highlights various
characteristics which are more general than the Cartesian representation. In these
systems the vectors are expressed by means of two kinds of components. Let the
representation of vector u in the plane coordinate system of oblique axes OXiXj that
1.3 Tensors 7
form an angle α, with basis vectors ei, ej (Fig. 1.4). The contravariant components
are obtained by means of straight lines parallel to the axes OXi and OXj and
graphed, respectively, as ui, uj (indicated with upper indexes). The covariant
componentsare obtained by means of projection on the axes OXi and OXj given,
respectively, by ui, uj (indicated with lower indexes).
The projection of vector u on an axis provides its component on this axis, and by
means of the dot product of u ¼ uiei and ej:
u � ej ¼ uiei � ej ¼ ui ei � ej
� � ¼ uiδij ¼ ui
that is the contravariant component of vector and in the same way by the covariant
component
u � ej ¼ uiei � ej ¼ ui ei � ej
� � ¼ uiδij ¼ ui
Thus, the vector is defined by its components
u ¼ uiei ¼ uiei
These components are not, in general, equal, and in the case of α ¼ 90o
(Cartesian coordinate systems), the equality of these components is verified.
j
j
u e
O O
j
X
j
X
i
X
i
X
j
X
j
e
j
u
i
X
j
e
i
e
i
u
ie
j
e
i
e
i
e
i
u
a b
Fig. 1.4 Vector components: (a) contravariant, (b) covariant
8 1 Review of Fundamental Topics About Tensors
1.3.3.4 Transformation Law of Covariant Vectors
The transformation law of base ei of an axis OX
i for a new axis OX
j
, with base ēj
(Fig. 1.5a), is given by
ej ¼ projei ej
�� ��� �ei ¼ 1 cos αð Þei
cos α ¼ ∂x
i
∂xj
ej ¼ ∂x
i
∂xj
ei
thatis the transformation law of the covariant basis. For the vector u the transfor-
mation of its covariant components is given by uj ¼ ∂xi∂xj ui, where the variables
relative to the original axis in relation to which the transformation performed are
found in the numerator of the equation.
1.3.3.5 Transformation Law of Contravariant Vectors
The projection of the vector ei on the axis OX
j (Fig. 1.5b) provides
ej ¼ projej eik k
� �
ei ¼ 1 cos αð Þei
cos α ¼ ∂x
j
∂xi
ej ¼ ∂x
j
∂xi
ei
that is the transformation law of the contravariant basis. For the vector u follows the
transformation law of its contravariant components uj ¼ ∂xj∂xi ui, where the variables
relative to the new axis, for which the transformation is carried out, are found in the
numerator of the expression.
ix∂ ix∂
O O
i
e
i
e
j
e
j
e
jx∂jx∂
ie
e
j
pro j
a b
Fig. 1.5 Transformation of coordinates: (a) covariant, (b) contravariant
1.3 Tensors 9
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1.3.4 Multilinear Forms
The tensors of the order p are multilinear forms, which are vectorial functions, and
linear in each variable considered separately. The concept of tensor is conceived by
means of the following approaches: (a) the tensor is a variety that obeys a trans-
formation law when changing the coordinate system; (b) this variety is invariant for
any coordinate system; and (c) there is an equivalence between these definitions
(equivalence law). A tensor of the order p is defined by a multilinear function with
Np components in the space EN, where R 1� N2
� � ¼ 0 represents its order, which is
maintained invariant if a change of the coordinate system occurs, and on the
rotation of the reference axes (linear and homogeneous transformation) its coordi-
nates modify according to a certain law.
Consider the space EN and the coordinate system X
i, i ¼ 1, 2, 3, . . .N, defined in
this space, where there are N equations that relate the coordinates of the points in
EN, given by continuously differentiable functions
xi ¼ xi xj� � i, j ¼ 1, 2, 3, . . .N ð1:3:4Þ
that transform these functions to a new coordinate systemX
i
. These transformations
of coordinates require only that N functions xi(xj) be independent. The necessary
and sufficient condition for this transformation to be possible is that J ¼ ∂xi
∂xj
			 			 6¼ 0.
The inverse function has an inverse Jacobian J ¼ ∂xj∂xi
			 			 and implies that JJ ¼ 1.
1.3.4.1 Transformation Law of the Second-Order Tensors
Let the position vector ui(x
i) expressed in the coordinate system Xi of base ei and a
new coordinate system X
i
, with same origin, in which the vector is expressed by
ui x
ið Þ. Consider the elements ∂xk
∂xi
of the rotation matrix that relates the coordinates of
these two systems, then follow by means of the transformation law of covariant
vectors
ui ¼ ∂x
k
∂xi
uk i, k ¼ 1, 2, 3 ð1:3:5Þ
vj ¼ ∂x
‘
∂xj
v‘ j, ‘ ¼ 1, 2, 3 ð1:3:6Þ
The vectors �ui(x
i) and vi x
ið Þ define the transformation of the second-order tensor
in terms of its covariant components
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Tij ¼ uivj ¼ ∂x
k
∂xi
∂x‘
∂xj
ukv‘ ¼ ∂x
k
∂xi
∂x‘
∂xj
Tk‘ ð1:3:7Þ
and for the contravariant components provides an analogous manner
T
ij ¼ uivj ¼ ∂x
i
∂xk
∂xj
∂x‘
ukv‘ ¼ ∂x
i
∂xk
∂xj
∂x‘
Tk‘ ð1:3:8Þ
In a same way, it follows for the transformation law in terms of the mixed
components
T
i
j ¼ uivj ¼
∂xi
∂xk
∂x‘
∂xj
ukv‘ ¼ ∂x
i
∂xk
∂x‘
∂xj
T k‘ ð1:3:9Þ
1.3.4.2 Transformation Law of the Third-Order Tensors
The transformations of the vectors u, v, w in terms of their covariant components
are given by
u‘ ¼ ∂x
i
∂x‘
ui vm ¼ ∂x
j
∂xm
vj wn ¼ ∂x
k
∂xn
wk
following by substitution
T‘mn ¼ u‘ vm wn ¼ ∂x
i
∂x‘
∂xj
∂xm
∂xk
∂xn
uivjwk
that leads to the following transformation law for the covariant components of the
third-order tensors
T‘mn ¼ ∂x
i
∂x‘
∂xj
∂xm
∂xk
∂xn
Tijk
and for the contravariant components
T
‘mn ¼ ∂x
‘
∂xi
∂xm
∂xj
∂xn
∂xk
Tijk
and in an analogous way, for the mixed components
T
mn
‘ ¼
∂x‘
∂xi
∂xm
∂xj
∂xn
∂xk
Tjki T
n
‘m ¼
∂x‘
∂xi
∂xm
∂xj
∂xn
∂xk
T kij
T
m
‘n ¼
∂x‘
∂xi
∂xm
∂xj
∂xn
∂xk
T jik T
‘m
n ¼
∂xi
∂x‘
∂xm
∂xj
∂xn
∂xk
Tijk
1.3 Tensors 11
1.3.4.3 Inverse Transformation
Let the inverse transformation of the vectors u and v of the coordinate system X
i
for
the coordinate system Xi, given by the covariant components of the vectors
ui ¼ ∂x
k
∂xi
uk vj ¼ ∂x
‘
∂xj
v‘ ð1:3:10Þ
It follows that
Tij ¼ uivj ¼ ∂x
k
∂xi
∂x‘
∂xj
ukv‘ ¼ ∂x
k
∂xi
∂x‘
∂xj
Tk‘ ð1:3:11Þ
Expression (1.3.11) allows concluding that a second-order tensor can be
interpreted as a transformation in the linear space E3, which associates the vector
u to the vector v by means of the tensorial product and that this linear and
homogeneous transformation has an inverse transformation. The inverse transfor-
mations are defined for the contravariant and mixed components in an analogous
way. Expressions (1.3.7) and (1.3.11) show that if the components of a second-
order tensor are null in a coordinates system, they will cancel each other in any
other coordinate system. For the definition of the transformation law of second-
order tensor to be valid, it is necessary that the transitive property apply to the linear
operators (Fig. 1.6).
1.3.4.4 Transitive Property
Let a second-order tensor Tk‘ defined in the coordinate system X
i, that is expressed
in the coordinate system X
i
by means of the expression (1.3.7), and with the
transformation of X
i
for eXi
iX
iX
iX�k
T
�
�
kji
k
ij Tx
x
x
x
T
¶
¶
¶
¶
=
ijq
j
p
i
pq Tx~
x
x~
x
T
~
¶
¶
¶
¶
=
�
�
kjp
k
pq Tx
x
x~
x
T
~
¶
¶
¶
¶
=
Fig. 1.6 Transitive
property of the second-order
tensors
12 1 Review of Fundamental Topics About Tensors
eTpq ¼ ∂xi∂exp ∂xj∂exq Tij ð1:3:12Þ
However, the tensor eTpq can be expressed in terms of tensor Tk‘, thereby avoiding
the intermediary transformation, so substituting expression (1.3.7) in expression
(1.3.12), it follows that
eTpq ¼ ∂xk
∂xi
∂x‘
∂xj
∂xi
∂exp ∂xj∂exq Tk‘ ð1:3:13Þ
and simplifying
∂xk
∂xi
∂xi
∂exp ¼ ∂xk∂exp ∂x‘∂xj ∂xj∂exq ¼ ∂x‘∂xj ð1:3:14Þ
Then
eTpq ¼ ∂xk∂exp ∂x‘∂xj Tk‘ ð1:3:15Þ
Expression (1.3.15) is the transformation law of the second-order tensor of the
coordinate system Xi for the coordinate system eXi, which proves that the transitive
property applies to these tensors. This property is also valid when using the
contravariant and mixed components.
The tensors studied in this book belong to metric spaces. If a variety is a tensor
with respect to the linear transformations, it will be a tensor with respect to all the
orthogonal linear transformations, but the inverse usually does not occur. The
tensors are produced in spaces more general than the vectorial space. Table 1.1
shows the covariant, contravariant, and mixed tensors and their transformation laws
for the space EN.
1.3.4.5 Multiplication of a Tensor by a Scalar
It is the multiplication that provides a new tensor as a result, which components are
the components of the original tensor multiplied by the scalar. Let the tensor Tijk
Table 1.1 Kinds of tensors
Tensor Expression Transformation law
Covariant Tij���k
Trs���t ¼ ∂x
i
∂xr
∂xj
∂xs
� � �∂x
k
∂xt
Tij���k
Contravariant Tij���k
T
rs���t ¼ ∂x
r
∂xi
∂xs
∂xj
� � � ∂x
t
∂xp
Tij���k
Mixed Tk‘���hij���f Tmn���hrs���t ¼
∂xi
∂xr
∂xj
∂xs
∂xm
∂xk
∂xn
∂x‘
� � �∂x
f
∂xt
∂xh
∂xh
Tk‘���hij���f
1.3 Tensors 13
and the scalar m which product Pijk is given by Pijk ¼ mTijk. For demonstrating this
expression represents a tensor, all that is needed is to apply the tensor transforma-
tion law to the same.
1.3.4.6 Addition and Subtraction of Tensors
The addition of tensors of the same order and the same type is given by
T kij ¼ Akij þ Bkij
The addition of the mixed tensors given by the previous expression provides as a
result a mixed tensor of the third order, which is twice covariant and oncecontravariant. To demonstrate this expression represents a tensor, all that is needed
is to apply the tensor transformation law to the same.
The subtraction is defined in the same way as the addition, however, admitting
that a tensor is multiplied by the scalar�1. As an exampleT kij ¼ Akij þ �1ð ÞBkij; thus,
this expression provides as a result a mixed tensor of the third order, which is twice
covariant and once contravariant.
To demonstrate that previous expression represents a tensor all that is needed is
to carry out the analysis developed for the addition considering the negative sign.
1.3.4.7 Contraction of Tensors
The contraction of a tensor is carried out when two of its indexes are made equal, a
covariant index and a contravariant index, and thus reducing the order of this tensor
in two. For instance, the tensor Tk‘ij contracted in the indexes ‘ and j results as
Tkji‘ ¼ Tkjij ¼ T ki .
1.3.4.8 Outer Product of Tensors
The outer product is the product of two tensors that provide as a new tensor, which
order is the sum of the order of these two tensors. Let, for example, the tensor Ak ...ij ...
with variance index number ( p, q) and the tensor B... ‘m... rs with variance index number
(u, v), which if multiplied provides a tensor Tk...‘mij...rs ¼ Ak...ij...B...‘m...rs with variance index
number pþ u, qþ vð Þ. The order of the tensor is given by the sum of these two
indexes. To demonstrate that the previous expression is a tensor, all that is needed is
to apply the tensor transformation law to the same.
14 1 Review of Fundamental Topics About Tensors
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1.3.4.9 Inner Product of Tensors
The inner product of two tensors is defined as the tensor obtained after the
contracting of the outer product of these tensors. Let, for example, tensors Aij and
B‘k which the outer product is P
‘
ijk ¼ AijB ‘k that provides as a result a tensor of the
fourth order, which contracted in the indexes ‘ and k provide the inner product
P ‘ij‘ ¼ AijB ‘‘ ¼ Pij. This shows that the resulting tensor is of the second order. To
demonstrate that this expression represents a tensor, all that is needed is to apply the
tensor transformation law to the same.
1.3.4.10 Quotient Law
This law allows verifying if a group of Np functions of the coordinates of the
referential system Xi has tensorial characteristics. Its application serves to test if a
variety is a tensor. The systematic for applying this law is to make the dot product of
the variety that is to be tested by a vector, for the outer product of two tensors
generates a tensor, and then carry out the contraction of this product and afterward,
by means of applying the tensor transformation law, verify if the variety fulfills
this law.
Let, for example, the contravariant tensor of the first order Tk and the variety
A(i, j, k) composed of 27 functions defined in the space EN, for which it is
desired to verify if it is tensor. The fundamental premise is that the vector Tk is
independent of A(i, j, k). If the inner product A i, j, kð ÞTk ¼ Bij originates a
contravariant tensor of the second order, then A(i, j, k) has the characteristics
of a tensor. Applying the transformation law of tensors to the tensor Bij
B
pq ¼ ∂x
p
∂xi
∂xq
∂xj
Bij ¼ ∂x
p
∂xi
∂xq
∂xj
A i; j; kð ÞTk
and for the vector Tk, it follows that
Tk ¼ ∂x
k
∂xr
T
r
By substitution
B
pq ¼ ∂x
p
∂xi
∂xq
∂xj
∂xk
∂xr
A i, j, kð ÞTr
and in a new coordinate system, the tensor Bij is given by
B
pq ¼ A p, q, rð ÞTr
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following by substitution
A p, q, rð Þ � ∂x
p
∂xi
∂xq
∂xj
∂xk
∂xr
A i, j, kð Þ
 �
T
r ¼ 0
As T
r
is an arbitrary vector the result is
A p; q; rð Þ ¼ ∂x
p
∂xi
∂xq
∂xj
∂xk
∂xr
A i; j; kð Þ
that represents the transformation law of third-order tensors. This shows that the
variety A(i, j, k) has tensorial characteristics.
1.4 Homogeneous Spaces and Isotropic Spaces
The isotropic space has properties which do not depend on the orientation being
considered, and the components of isotropic tensors do not change on an orthogonal
linear transformation. The sum of isotropic tensors results in an isotropic tensor,
and the product of isotropic tensors is also an isotropic tensor.
There is no isotropic tensor of the first order. The isotropic tensor of the fourth
order is given by
Tijk‘ ¼ λδijδk‘ þ μδikδj‘ þ νδi‘δjk ð1:4:1Þ
where λ, μ, ν are scalars. The Kronecker delta δij is the only isotropic tensor of the
second order.
The homogeneous space has properties which are independent of the position of
the point. The homogeneous tensors have constant components when the coordinate
system is changed.
A homogeneous tensor of the fourth order is given by
Tij‘k ¼ λδijδk‘ þ μ δikδj‘ þ δi‘δjk
� � ð1:4:2Þ
where λ, μ are scalars.
1.5 Metric Tensor
The study of tensors carried out in affine spaces applies to another type, called
metric space, in which the length of the curves is determined by means of a variety
that defines this space, in which the basic magnitudes are the length of a curve and
the vector’s norm, just as the angle between vectors and the angle between two
curves. The distinction between these two types of spaces is of fundamental
importance in the study of tensors.
16 1 Review of Fundamental Topics About Tensors
The metric space is determined by the definition of its fundamental tensor which
is related with its intrinsic properties. The conception of this metric tensor, which
gives an arithmetic form to the space, considers the invariance of distance between
two points, the concept of distance being acquired from the space E3. The geometry
grounded in the concept of metric tensor is called Riemann geometry.
The angle between two curves is calculated by means of the dot product between
vectors using the metric tensor, which awards a generalization to this tensor’s
formulation. Let the arc element length of a curve ds defined in the Cartesian
coordinate system Xi with unit vectors g1, g2, g3 by means of its coordinates x
i
(Fig. 1.7), with two neighboring pointsP xið Þ , Q xi þ dxið Þ, which define the position
vectors r and rþ dr, respectively. The coordinates of increment of the position
vector dr are given byQ� P ¼ dxi; thus, lim
Q!P
Q� Pð Þ ¼ ds, and the dot product of
this vector by himself takes the form
ds2 ¼ dr � dr ¼ dxidxi ð1:5:1Þ
Consider a transformation of the coordinates xi ¼ xi xið Þ for a new coordinate
system X
i
dxi ¼ ∂x
i
∂xk
dxk ð1:5:2Þ
becomes
ds2 ¼ ∂x
i
∂xk
∂xi
∂x‘
dxkdx‘ ð1:5:3Þ
Putting
gk‘ ¼
∂xi
∂xk
∂xi
∂x‘
ð1:5:4Þ
2
X
1
X
3
X
ii
d xxQ +
i
xP
d s
O
1
g
2
g
3
g
Fig. 1.7 Elementary arc
of a curve
1.5 Metric Tensor 17
thus the metric takes the form
ds2 ¼ gk‘dxkdx‘ ð1:5:5Þ
The symmetry of the variety given by expression (1.5.4) is obvious, because
gk‘ ¼ g‘k, then
gij ¼ gji ¼
g1g1 g1g2 g1g3
g2g1 g2g2 g2g3
g3g1 g3g2 g3g3
24 35 ¼ g11 g12 g13g21 g22 g23
g31 g32 g33
24 35 ð1:5:6Þ
The analysis of expression (1.5.4) shows that gk‘ relates with the Jacobian
J½ � ¼ ∂xi
∂xk
h i
of a linear transformation by means of the following expression
gk‘½ � ¼
∂xi
∂xk
 �T
∂xi
∂x‘
 �
¼ J½ �T J½ � ð1:5:7Þ
For the coordinate system Xi, the variety gij is defined by his unit vectors gi, gj.
Consider a new coordinate system X
i
, with respect to which these unit vectors are
expressed by
gk ¼
∂xi
∂xk
gi g‘ ¼
∂xj
∂x‘
gj ð1:5:8Þ
Thus
gk‘ ¼
∂xi
∂xk
gi
� �
∂xj
∂x‘
gj
� �
¼ ∂x
i
∂xk
∂xj
∂x‘
gigj
� � ¼ ∂xi
∂xk
∂xj
∂x‘
gij
then gij is a symmetric tensor of the second order.
The arc length is invariable when changing the coordinate system. The coeffi-
cients of gk‘(x
i) are class C2, and the N equations xi ¼ xi xið Þ must satisfy the
1
2
N N þ 1ð Þ partial differential equations given by expression (1.5.4). However, if
gk‘(x
i) is specified arbitrarily, this system of 1
2
N N þ 1ð Þ partial differential equa-
tions, in general, has no solution. The fundamental tensor gk‘ related toa coordinate
system Xi, in a region of the space EN, must fulfill the following conditions:
(a) gk‘(x
i) is a class C2 function, i.e., its second-order derivatives exist and are
continuous.
(b) Be symmetrical, i.e., gk‘ ¼ g‘k.
(c) detgk‘ ¼ g 6¼ 0, i.e., gk‘ is not singular.
(d) ds2 ¼ gk‘dxkdx‘ is an invariant after a change of coordinate system.
18 1 Review of Fundamental Topics About Tensors
Expression (1.5.5) is put under parametric form with the coordinates xi ¼ xi tð Þ
and i ¼ 1, 2, 3 . . .N, and the parameter a 	 t 	 b provides
s ¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gk‘
dxk
dt
dx‘
dt
���� ����
s
dt ð1:5:9Þ
Admit a functional parameter hi ¼ 
1, so as to allow the conditions gk‘ dx
k
dt
dx‘
dt > 0 and gk‘
dxk
dt
dx‘
dt < 0 to be be used instead of the absolute value shown in
expression (1.5.9), because the use of hi is more adequate to the algebraic
manipulations; thus,
s ¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higk‘
dxk
dt
dx‘
dt
r
dt ð1:5:10Þ
The quadratic and homogeneous form Φ ¼ gk‘dxkdx‘ is called metric or funda-
mental form of the space, being invariant after a change of coordinate system. In
space E3 with Φ > 0, which provides g > 0, and when Φ ¼ 0, the initial and final
points of the arc coincide. If Φ ¼ 0 and dxi are not all null, the displacement
between the two points is null. The possibility of Φ being undefined is admitted, for
instance, in the case Φ ¼ dx1ð Þ 2 � dx2ð Þ 2, for which dx1 ¼ dx2 results in Φ ¼ 0.
This case is interpreted as having a null displacement of the point. If dxi 6¼ 0, i.e.,
the displacements are not null, hi is adopted so that hiΦ > 0.
The spaces EN (hyperspaces) are analyzed in an analogous way to the analysis of
the space E3 by means of defining a metric, formalizing the Riemann geometry. The
geometries not grounded on the concept of metric are called non-Riemann
geometries.
To demonstrate that expression (1.5.10) is invariant through a change in its
parametric representation, let a curve of class C2 represented by means of the
coordinates xi ¼ xi tð Þ and a 	 t 	 b. Consider a transformation for the new
coordinates xi ¼ xi tð Þ and a 	 t 	 b, where t ¼ f tð Þ with f 0 tð Þ > 0, and in the
new limits a ¼ f að Þ, b ¼ f bð Þ. Applying the chain rule to the function t ¼ f tð Þ:
dt
dt
¼ f 0 tð Þ ) dt ¼ dt
f
0
tð Þ ð1:5:11Þ
and with expression (1.5.11) in expression (1.5.10)
1.5 Metric Tensor 19
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L¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higij
dxi
dt
dxj
dt
r
dt ¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higij
dxi
dt
dxj
dt
f
0
tð Þ
h i2r
dt ¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higij
dxi
dt
dxj
dt
r
f
0
tð Þdt
¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higij
dxi
dt
dxj
dt
r
dt ¼ L
then the value of this expression does not vary with the change of the curve’s
parameterization.
The metric can be written in matrix form so as to make the usual calculations
easier
ds
dt
� �2
¼ dx
k
dt

 �T
gij
h i dx‘
dt

 �
ð1:5:12Þ
In the space E3, the metric is defined by
dt
dt
¼ f 0 tð Þ ) dt ¼ dt
f
0
tð Þ ð1:5:13Þ
ds2 ¼ g11dx1dx1 þ g12dx1dx2 þ g13dx1dx3
þ g21dx2dx1 þ g22dx2dx2 þ g23dx2dx3
þ g31dx3dx1 þ g32dx3dx2 þ g33dx3dx3
ð1:5:14Þ
or
ds2 ¼ gii dxi
� �2 þ gkk dxk� �2 þ 2gikdxidxk ð1:5:15Þ
For the particular case in which the coordinate systems are orthogonal (Fig. 1.8),
the segments on the coordinate axes Xi are defined by the unit vectors gi of these
axes
ds ið Þ ¼ gidxi ð1:5:16Þ
which provide the metric
ds2 ¼ h1g1dx1
� �2 þ h2g2dx2� �2 þ h3g3dx3� �2
¼ h1dx1
� �2 þ h2dx2� �2 þ h3dx3� �2 ð1:5:17Þ
then the metric tensor is defined by the elements of the diagonal of the matrix
20 1 Review of Fundamental Topics About Tensors
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gij ¼
h21 0 0
0 h22 0
0 0 h23
24 35 ð1:5:18Þ
where h1 ¼ ffiffiffiffiffiffig11p , h2 ¼ ffiffiffiffiffiffig22p , h3 ¼ ffiffiffiffiffiffig33p , and detgij ¼ g ¼ g11g22g33.
Exercise 1.1 Let gijx
ixj ¼ 0, 8xi, xj show that gk‘ þ g‘k ¼ 0.
Putting
Φ ¼ gijxixj ¼ 0
and differentiating with respect to xk
∂Φ
∂xk
¼ gij
∂xi
∂xk
xj þ gijxi
∂xj
∂xk
¼ gijδ ikxj þ gijδ jkxi ¼ gkjxj þ gikxi ¼ 0
Differentiating with respect to x‘
∂2Φ
∂xk∂x‘
¼ gkj
∂xj
∂x‘
þ gik
∂xi
∂x‘
¼ gkjδ j‘ þ gikδ i‘ ¼ gk‘ þ g‘k ¼ 0 Q:E:D:
Exercise 1.2 Calculate the length of the curve of class C2 given by the parametric
equations x1¼ 3� t, x2 ¼ 6tþ 3, and x3 ¼ ‘n t, in the space defined by the metric
tensor
1
g
2
g
3
g
1X
2X
3X
O
Fig. 1.8 Orthogonal
coordinate systems
1.5 Metric Tensor 21
gij ¼
12 4 0
4 1 1
0 1 x1ð Þ2
24 35
The metric of the space in matrix form stays
ds
dt
� �2
¼ dx
k
dt

 �T
gij
h i dx‘
dt

 �
and with the derivatives
dx1
dt
¼ �1 dx
2
dt
¼ 6 dx
3
dt
¼ 1
t
it follows
ds
dt
� �2
¼ �1; 6; 1
t

 � 12 4 0
4 1 1
0 1 3� tð Þ2
24 35 �16
1
t
8><>:
9>=>; ¼ tþ 3ð Þ
2
t2
Making h1 ¼ 1 in expression
s ¼
ðb
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higk‘
dxk
dt
dx‘
dt
r
dt ) L ¼
ðe
1
tþ 3
t
� �
dt ¼ eþ 2
1.5.1 Conjugated Tensor
Let the increment of the position vector expressed by means of their covariant
components dr ¼ rigi in any coordinate system, where gj is the basis vector of this
referential and with the dot product
dr � dr ¼ dxidxj gigj
� � ð1:5:19Þ
and
gigj
� � ¼ gij ¼ gji ð1:5:20Þ
whose symmetry comes from the commutative property of the dot product.
22 1 Review of Fundamental Topics About Tensors
This variety with properties analogous to the properties of the metric tensor is
represented by nine components of a symmetrical matrix 3� 3, which form a
second-order contravariant tensor. It is called conjugated metric tensor; thus,
gij ¼
g11 g12 g13
g21 g22 g23
g31 g32 g33
24 35 ð1:5:21Þ
The definition of the conjugated of the metric tensor is given by
gij ¼ gigj ð1:5:22Þ
and with the relations between the reciprocal basis
gi ¼ gk � g‘
V
gj ¼ gm � gn
V
ð1:5:23Þ
results for the conjugated metric tensor
gij ¼ 1
V2
gk � g‘ð Þ gm � gnð Þ ð1:5:24Þ
but with the fundamental formula of the vectorial algebra
gk � g‘ð Þ � gm � gnð Þ ¼ gk � g‘ð Þ � gm½ � � gn ð1:5:25Þ
and developing the double-cross product in brackets
gk � g‘ð Þ � gm½ � � gn ¼ gk � gmð Þg‘ � g‘ � gmð Þgk ð1:5:26Þ
So
gij ¼ 1
V2
gk � gmð Þ g‘ � gnð Þ � g‘ � gmð Þ gk � gnð Þ½ � ð1:5:27Þ
The term in brackets in expression (1.5.27) is the development of the
determinant
Gij ¼ gk � gm gk � gn
g‘ � gm g‘ � gn
				 				 ¼ gkm gkng‘m g‘n
				 				 ð1:5:28Þ
Then
gij ¼ Gij
V2
ð1:5:29Þ
1.5 Metric Tensor 23
Summarizing these analyses by means of the transcription of the following
expressions
gij ¼
Gij
V
2
¼ Gij
g
gij ¼ G
ij
V2
¼ G
ij
g
ð1:5:30Þ
Thus
V ¼ 
 ffiffiffigp ¼ 
 ffiffiffiffiffiffiffiffiffiffiffidetgijp V ¼ 
 ffiffiffigp ¼ 
 ffiffiffiffiffiffiffiffiffiffiffiffidetgijq ð1:5:31Þ
The sign þð Þ in expressions (1.5.31) corresponds to a levorotatory coordinates,
and the sign �ð Þcorresponds to a dextrorotatory coordinates. Knowing thatVV ¼ 1,
it follows that gg ¼ 1.
Exercise 1.3 Let detgij x
nð Þ ¼ g xnð Þ. Calculate the derivative ∂g∂xn, n ¼ 1, 2, . . ..
The matrix linked to the determinant g is a function of the variables xn:
gij ¼ gij xi
� �
and this determinant being a function of the matrix elements
g ¼ g gij
� �
by the chain rule
∂g
∂xn
¼ ∂g
∂gij
∂gij
∂xn
As det g is expressed by its cofactors
g ¼ g1kGk1 ¼ g11G11 þ g12G21 þ g13G31 þ � � �
and the terms Gk1 do not contain the terms g1k, so
∂g
∂g11
¼ G11 ∂g∂g12
¼ G21 ∂g∂g13
¼ G31 � � �
Generalizing provides
∂g
∂gij
¼ Gji
24 1 Review of Fundamental Topics About Tensors
By substitution
∂g
∂xn
¼ Gji
∂gij
∂xn
Exercise 1.4 Calculate the derivative of detg ¼ x
1x2 x1ð Þ2
x1ðÞ2 2x1
					
					with respect to the
variable x1.
From Exercise 1.3
∂g
∂xi
¼ Gji
∂gij
∂xi
This expression is the sum of n determinants. Each of these determinants differs
from the determinant g only in the lines and columnswhich are being differentiated, so
∂g
∂x1
¼
∂g11
∂x1
∂g12
∂x1
g21 g22
					
					þ
g11 g12
∂g21
∂x1
∂g22
∂x1
						
						 ¼ x
2 2x1
x1ð Þ 2 2x1
				 				þ x1x2 x1ð Þ 2
2x1 2
					
					
Exercise 1.5 Let g ¼ detgij the determinant of the metric tensor gij and xk an
arbitrary variable. Calculate (a) ∂ ‘n gð Þ∂gij and (b)
∂ ‘n gð Þ
∂xk .
(a) From Exercise 1.3
∂g
∂gij
¼ Gji
but as gij ¼ gji it follows that
∂g
∂gji
¼ Gij
Expression (1.5.30) provides
gij ¼ G
ij
g
gij ¼
Gij
g
) Gij ¼ ggij
By substitution
∂g
∂gji
¼ ggij ¼ ggij
1.5 Metric Tensor 25
whereby
∂ ‘ngð Þ
∂gij
¼ 1
g
∂g
∂gij
) ∂ ‘ngð Þ
∂gij
¼ gij
(b) By the chain rule
∂ ‘ngð Þ
∂xk
¼ ∂ ‘ngð Þ
∂gij
∂gij
∂xk
and substituting the result obtained in the previous item in this expression
∂ ‘ngð Þ
∂xk
¼ gij ∂gij
∂xk
Exercise 1.6 Calculate the metric tensor, its conjugated tensor, and the metric for
the Cartesian coordinate system.
Let the Cartesian coordinates (x1, x2, x3), and by the definition of the distance
between two points
ds2 ¼ dx1� �2 þ dx2� �2 þ dx3� �2
which is the square of the metric, thus
ds2 ¼ δijdxidxj
By the definition of the metric tensor and the conjugated metric tensor, then
gij ¼ δij ¼
1 0 0
0 1 0
0 0 1
24 35
gij ¼ 1
gij
¼
1 0 0
0 1 0
0 0 1
24 35
Exercise 1.7 Calculate the metric tensor, its conjugated tensor, and the metric for
the cylindrical coordinate system given by r � x1, θ � x2; and z � x3 where
�1 	 r 	 1, 0 	 θ 	 2π, and �1 	 z 	 1, which relations with the Cartesian
coordinates are x1 ¼ x1 cos x2, x2 ¼ x1 sin x2, and x3 � x3.
26 1 Review of Fundamental Topics About Tensors
With the definition of metric tensor
gij ¼
∂xk
∂xi
∂xk
∂xj
¼ ∂x
1
∂xi
∂x1
∂xj
þ ∂x
2
∂xi
∂x2
∂xj
þ ∂x
3
∂xi
∂x3
∂xj
– i ¼ j ¼ 1
g11 ¼
∂x1
∂x1
∂x1
∂x1
þ ∂x
2
∂x1
∂x2
∂x1
þ ∂x
3
∂x1
∂x3
∂x1
¼ cos x2� �2 þ sin x2� �2 þ 0 ¼ 1
– i ¼ j ¼ 2
g22 ¼
∂x1
∂x2
∂x1
∂x2
þ ∂x
2
∂x2
∂x2
∂x2
þ ∂x
3
∂x2
∂x3
∂x2
¼ �x1 sin x2� �2 þ x1 cos x2� �2 þ 0 ¼ x1� �2
– i ¼ j ¼ 3
g33 ¼
∂x1
∂x3
∂x1
∂x3
þ ∂x
2
∂x3
∂x2
∂x3
þ ∂x
3
∂x3
∂x3
∂x3
¼ 0þ 0þ 1 ¼ 1
– i ¼ 1 , j ¼ 2
g12 ¼
∂x1
∂x1
∂x1
∂x2
þ ∂x
2
∂x1
∂x2
∂x2
þ ∂x
3
∂x1
∂x3
∂x2
¼ cos x2 �x1 sin x2� �2 þ sin x2 x1 cos x2� �2 þ 0 ¼ 0
For the other terms g21 ¼ g13 ¼ g31 ¼ g23 ¼ g32 ¼ 0, then
ds2 ¼ g11dx1dx1 þ g22dx2dx2 þ g11dx2dx2 ¼ dx1
� �2 þ x1� �2 dx2� �2 þ dx3� �2
¼ drð Þ2 þ rdθð Þ2 þ dzð Þ2
The metric tensor and its conjugated tensor are given, respectively, by
gij ¼
1 0 0
0 r2 0
0 0 1
264
375 gij ¼ 1
gij
¼
1 0 0
0
1
r2
0
0 0 1
264
375
and with the base vectors
gi ¼
∂xj
∂xi
ej
1.5 Metric Tensor 27
i ¼ 1 ) g1 ¼
∂xj
∂x1
ej
j ¼ 1, 2, 3 ) g1 ¼
∂x1
∂x1
e1 þ ∂x
2
∂x1
e2
∂x3
∂x1
e3
g1 ¼ cos x2e1 þ sin x2e2
8>>>>><>>>>>:
i ¼ 2 ) g2 ¼
∂xj
∂x2
ej
j ¼ 1, 2, 3 ) g2 ¼
∂x1
∂x2
e1 þ ∂x
2
∂x2
e2
∂x3
∂x2
e3
g2 ¼ �x1 sin x2e1 þ x1 cos x2e2
8>>>>><>>>>>:
i ¼ 3 ) g3 ¼
∂xj
∂x3
ej
j ¼ 1, 2, 3 ) g3 ¼
∂x1
∂x3
e1 þ ∂x
2
∂x3
e2
∂x3
∂x3
e3
g3 ¼ 0þ 0þ 1 � e3 ¼ e3
8>>>>><>>>>>:
By means of the dot products
gi � gj ¼ δij ei � ej ¼ δij
it follows for the components of the metric tensor
g11 ¼ g1 � g1 ¼ cos x2e1 þ sin x2e2
� � � cos x2e1 þ sin x2e2� � ¼ 1
g22 ¼ g2 � g2 ¼ �x1 sin x2e1 þ x1 cos x2e2
� � � �x1 sin x2e1 þ x1 cos x2e2� � ¼ x2� �2
g33 ¼ g3 � g3 ¼ e3ð Þ � e3ð Þ ¼ 1
The other components of this tensor are null.
Exercise 1.8 Calculate the metric tensor, its conjugated tensor, and the metric for
the spherical coordinate system r � x1,φ � x2, θ � x3, �1 	 r 	 1, and
0 	 φ 	 π, where 0 	 θ 	 2π, which relations with the Cartesian coordinates are
x1 ¼ x1 sin x2 cos x3, x2 ¼ x1 sin x2 sin x3,and x3 � x1 cos x2.
With the definition of metric tensor
gij ¼
∂xk
∂xi
∂xk
∂xj
) gij ¼
∂x1
∂xi
∂x1
∂xj
þ ∂x
2
∂xi
∂x2
∂xj
þ ∂x
3
∂xi
∂x3
∂xj
– i ¼ j ¼ 1
28 1 Review of Fundamental Topics About Tensors
g11 ¼
∂x1
∂x1
∂x1
∂x1
þ ∂x
2
∂x1
∂x2
∂x1
þ ∂x
3
∂x1
∂x3
∂x1
¼ sin x2 cos x3� � 2 þ sin x2 sin x3� � 2 þ cos x2� � 2 ¼ 1
– i ¼ j ¼ 2
g22 ¼
∂x1
∂x2
∂x1
∂x2
þ ∂x
2
∂x2
∂x2
∂x2
þ ∂x
3
∂x2
∂x3
∂x2
¼ x1 cos x2 cos x3� � 2 þ x1 cos x2 sin x3� � 2 þ �x1 sin x2� � 2 ¼ x1� � 2
– i ¼ j ¼ 3
g33 ¼
∂x1
∂x3
∂x1
∂x3
þ ∂x
2
∂x3
∂x2
∂x3
þ ∂x
3
∂x3
∂x3
∂x3
¼ �x1 sin x2 sin x3� �2 þ x1 sin x2 cos x3� �2 þ 0 ¼ x1 sin x2� �2
For the other terms g12 ¼ g21 ¼ g13 ¼ g31 ¼ g23 ¼ g32 ¼ 0, then
ds2 ¼ g11dx1dx1 þ g22dx2dx2 þ g33dx3dx3 ¼ dx1
� �2 þ x3� �2 dx2� �2 þ x1 sin x2� �2 dx3� �2
¼ drð Þ2 þ rdφð Þ2 þ r sin θdθð Þ2
The metric tensor and its conjugated tensor are given, respectively, by
gij ¼
1 0 0
0 r2 0
0 0 r2 sin 2φ
264
375 gij ¼ 1
gij
¼
1 0 0
0
1
r2
0
0 0
1
r2 sin 2φ
266664
377775
Exercise 1.9 Calculate the metric tensor, its conjugated tensor, and the metric for
the cylindrical elliptical coordinate system ξ � x1, η � x2, and z � x3, where ξ � 0,
0 	 η 	 2π, �1 	 z 	 1, which relations with the Cartesian coordinates are
x1 ¼ coshx2 cos x2, x2 ¼ sinhx2 sin x2, x3 � x3.
With x3 ¼ const:, the elliptical cylinder is x10 ¼ const::
x1
chx10
� �2
þ x
2
shx10
� �2
¼ cos x2� �2 þ sin x2� �2 ¼ 1
dx1 ¼ sinhx1 cos x2dx1 dx2 ¼ coshx1 sin x2dx1
ds¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dx1ð Þ2 þ dx2ð Þ2
q
¼ cosh2x1 � cos 2x2� �dx1
g11 ¼ cosh2x1 � cos 2x2
� �
1.5 Metric Tensor 29
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With x1 ¼ const: the hyperbolic cylinder is x20 ¼ const::
x1
cos x20
� �2
� x
2
sin x20
� �2
¼ coshx1� � 2 � sinhx1� � 2 ¼ 1
dx1 ¼ �coshx1 sin x2dx2 dx2 ¼ sinhx1 cos x2dx2
ds¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dx1ð Þ2 þ dx2ð Þ2
q
¼ cosh2x1 � cos 2x2� �dx2
g22 ¼ cosh2x1 � cos 2x2
� �
For x3 � x3 provides dx2 ¼ dx3, whereby g33 ¼ 1, following
ds2 ¼ cosh2x1 � cos 2x2� �2 dx1� �2 þ cosh2x1 � cos 2x2� �2 dx2� �2 þ dx3� �2
The metric tensor and its conjugated tensor are given, respectively, by
gij ¼
cosh2x1 � cos 2x2 0 0
0 cosh2x1 � cos 2x2 0
0 0 1
264
375
gij ¼
1
cosh2x1 � cos 2x2 0 0
0
1
cosh2x1 � cos 2x2 0
0 0 1
26664
37775
1.5.2 Dot Product in Metric Spaces
Let the vectors u and v contained in the metric space EN defined by the fundamental
tensor gk‘. The dot product u � v with u ¼ uiei and v ¼ vjej depends only on the
vectors and is independent of the coordinate system in relation to which the same is
specified. It is observed that only when the coordinates of the vectors are covariant
and contravariant, this product is like to the dot product in Cartesian coordinates.
The dot product is invariant in view of the transformation of coordinates
u � v¼ uiei � vjej ¼ gijuivj ¼ uiei:vjej ¼ gijuivj ¼ uiei:vjej
¼ gji uivj ¼ uivj ¼ uivj
ð1:5:32Þ
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1.5.2.1 Vector Norm
The generalization of the dot product of vectors for a metric space EN allows
obtaining the norm of a vector. Let vector v with norm (modulus)
vk k ¼ ffiffiffiffiffiffiffiffiv � vp ¼ ffiffiffiffiffiv2p
that is equal to the distance between the extreme points, thus, with the expression of
the metrics
v2 ¼ higk‘vkv‘
results for the norm of the vector in terms of its contravariant components
vk k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higkkv
kvk
q
In an analogous way for the covariant components vk
vk k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
higkkvkvk
p
and for Cartesian coordinates
vk k ¼
ffiffiffiffiffiffiffiffi
vkvk
p
If v is a unit vector, the expressions provide
higkkv
kvk ¼ 1 higkkvkvk ¼ 1 vkvk ¼ 1
The properties of the vectors norm are:
(a) vk k � 0, which is a trivial property, for the norm will only cancel itself if v
is null.
(b) mvk k ¼ mk k vk k, where m is a scalar.
(c) uþ vk kuk k þ vk k.
(d) u � vk k 	 uk k � vk k, Cauchy–Schwarz inequality.
For the case of non-null vectors, the equality of the relation (d) exists only if
u ¼ mv, where m is a scalar.
Exercise 1.10 Calculate the modulus of vector u(1; 1; 0; 2) in space E4, defined by
the metric tensor
gij ¼
�1 0 0 0
0 �1 0 0
0 0 �1 0
0 0 0 c2
2664
3775
1.5 Metric Tensor 31
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For the line element ds2 ¼ gijuiuj, and developing this expression
ds2 ¼ gijuiuj ¼ g11u1u1 þ g22u2u2 þ g33u3u3 þ g44u4u4
¼ �1� 1� 1� 1� 1� 1þ 0þ c2 � 2� 2 ¼ �2þ 4c2
ds ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2c2 � 1ð Þ
p
1.5.2.2 Lowering of a Tensor’s Indexes
By means of analysis referent to the transformation of the covariant components of
the vector in their contravariant components, and vice versa, it is verified that inner
product of a tensor by the metric tensor allows raising or lowering the indexes of
this tensor.
For multiplying the contravariant tensor of the first order, i.e., the contravariant
vector Ti by the tensor gk‘, results in T
i
k‘ ¼ gk‘Ti, and for the contraction i ¼ ‘, then
T iki ¼ gkiTi ¼ Tk that is a covariant vector. The index of the original vector was
lowered and its order reduced in two units.
1.5.2.3 Raising of a Tensor’s Indexes
Let the covariant vector Tk, which multiplied by g
ik, provides as a result the tensor g
ikTk, and changing the covariant coordinates of the vector by its contravariant
coordinates
gikTk ¼ gik gk‘T‘
� � ¼ δ i‘T‘ ¼ Ti
that is a contravariant vector. The index of the original vector was raised. Then a
covariant vector is obtained by means of the inner product of a contravariant vector,
this indexes transformation process as being reciprocal. The vectors Ti and Ti are
called associated vectors, and it refers to the contravariant and covariant compo-
nents of the vector.
For the case of second-order tensors, an analysis is carried out that is analogous
to the one developed for the vectors. Let the covariant tensor of the second-order Tk‘
and its associated tensor Tij ¼ gikgj‘Tk‘.
It is verified in the general case that these tensors are not conjugated tensors,
for example, when Tij ¼ mgij, where m is a scalar, the tensor Tk‘ will be a multiple
of gk‘,
Tk‘ ¼ gikgj‘Tij ¼ gikgj‘ mgij
� �
¼ mgikδ ‘i ¼ mg‘k ¼ mgk‘
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The raising and lowering operations of the indexes of tensors are carried out
adopting, firstly, a point for indicating where the position to be left empty in the
index that will be raised or lowered. For example, for the tensor Tji, the empty
position is indicated by means of the notation Tj
 i, and in an equal manner A
rs

 
 p
exists for the tensor Arsp .
Let the inner product of the tensor Ti
 jk by the metric tensor g‘i, g‘iT
i

jk ¼ T‘jk in
which the upper index was lowered, and gijT
kj ¼ T k
i that had an index lowered, or
further, gijgk‘T
j‘ ¼ Tik, which two upper indexes were lowered.
For raising the indexes, in an analogous way to the raising of an index gijTjk
¼ T i
k or gkjTij ¼ T
ki , thus gijgk‘Tj‘ ¼ Tik.
In the case in which the index is lowered and then raised, the original tensor is
obtained gkjT
ij ¼ T i
k and next gkjT i
k ¼ Tij.
1.5.2.4 Tensorial Equation
If a term of a tensorial equation contains a dummy index, it can be raised or
lowered, i.e., change the position without changing the value of the equation. The
following example illustrates this assertion
Ai
jBi ¼ gkiAkj
� �
gi‘B
‘
� � ¼ gkigi‘AkjB‘ ¼ δ k‘ AkjB‘ ¼ A‘jB‘ ¼ AijBi
where the index i was lowered in one tensor and raised in the tensor.
If a free index is a part of the tensorial expression, a new tensorial expression
equivalent to this one can be obtained, lowering or raising this index in the members
of the original expression. To illustrate this assertion, the following tensorial
equation is admitted
Tijk ¼ AijBk
which is equivalent to
gi‘T‘jk ¼ gi‘A‘jBk
so it results in
T i
jk ¼ Ai
jBk
where the index i was raised.
1.5 Metric Tensor 33
Exercise 1.11 Raise and lower the indexes of vector u, for the metric tensor and its
conjugated tensor:
(a) uj ¼
3
4
5
8<:
9=; gij ¼
1 0 0
0 x1ð Þ2 0
0 0 x1 sin x2ð Þ2
24 35
(b) uj ¼
5
4
3
8<:
9=; gij ¼
1 0 0
0 x1ð Þ�2 0
0 0 x1 sin x2ð Þ�2
24 35
(a) Carrying out the following matrix multiplication provides the covariant com-
ponents of the vector
ui ¼ gijuj ¼
1 0 0
0 x1ð Þ2 0
0 0 x1 sin x2ð Þ2
24 35 34
5
8<:
9=; ¼
3
4 x1ð Þ2
5 x1 sin x2ð Þ2
8<:
9=;
(b) Carrying out the following matrix multiplication provides the contravariant
components of the vector
ui ¼ gijuj ¼
1 0 0
0 x1ð Þ�2 0
0 0 x1 sin x2ð Þ�2
24 35 54
3
8<:
9=; ¼
5
4 x1ð Þ�2
3 x1 sin x2ð Þ�2
8<:
9=;
Exercise 1.12 Given the covariant basis g1 ¼ e1; g2 ¼ e1 þ e2; g3 ¼ e3 and the
tensor of the space gij ¼
1 1 1
1 2 2
1 2 3
24 35, calculate the vectors of the contravariant
basis and the conjugated metric tensor.
The determinant of the metric tensor is given by
g ¼ detgi ¼
1 1 1
1 2 2
1 2 3
						
						 ¼ 1
which indicates that the system is dextrorotary.
For the vectors of the contravariant basis, it follows that
g1 ¼ g2 � g3
g
¼
e1 e2 e3
1 1 0
1 1 1
						
						 ¼ e1 � e2
g2 ¼ g3 � g1
g
¼
e1 e2 e3
1 1 1
1 0 0
						
						 ¼ e2 � e3
34 1 Review of Fundamental Topics About Tensors
g3 ¼ g1 � g2
g
¼
e1 e2 e3
1 0 0
1 1 0
						
						 ¼ e3
then the conjugated metric tensor is given by
gij ¼ gi � gj ¼
2 �1 0
�1 2 �1
0 �1 1
24 35
The verification of the operation is carried out by means of the expression
gijg
ij ¼ δ ij , thus
1 1 1
1 2 2
1 2 3
24 35 2 �1 0�1 2 �1
0 �1 1
24 35 ¼ 1 0 00 1 0
0 0 1
24 35
1.5.2.5 Associated Tensors
The metric tensor gij and its conjugated tensor g
ij relate intrinsically to each other,
which allows using them for analyzing the relations between the covariant and
contravariant components of the vector u
ui ¼ gijuj ui ¼ gijuj ð1:5:33Þ
These expressions generate two linear equation systems, which unknown quan-
tities are u1, u2, u3, and u
1, u2, u3. The solution of the system given by means of
Cramer’s rule and with the determinant of the metric tensor
g ¼ detgij ¼
g11 g12 g13
g21 g22 g23
g31 g32 g33
						
						
and its cofactor
Gij ¼ gkm gkn
g‘m g‘n
				 				
thus
ui ¼ gijuj ¼
Giju
j
g
) gij ¼
Gij
g
1.5 Metric Tensor 35
In an analogous way for the system given by expression 1.5.33
ui ¼ gijuj ¼ G
ijuj
g
) gij ¼ G
ijuj
g
The linear operators gij, g
ij allow relating the covariant and contravariant com-
ponents of vector u. Defining this vector by means of its covariant components and
performing the dot product of this vector by the basis unit vectors gi:
u � gi ¼ ujgj
� � � gi ¼ uj gj � gi� � ¼ gijuj ¼ ui
that are the contravariant components of vector u.
In an analogous way for the transformation of the contravariant components in
the covariant components
u � gi ¼ uigj
� � � gi ¼ ui gj � gi� � ¼ gijuj ¼ ui
These expressions relate to each other in a kind of coordinate as a function of the
other, where the tensors gij, g
ij are the operators responsible for these
transformations.
The covariant components and the contravariant components of the vector u are
given, respectively, by
ui ¼
u1 ¼ g11u1
u2 ¼ g22u2
u3 ¼ g33u3
8<: ui ¼ u
1 ¼ g11u1
u2 ¼ g22u2
u3 ¼ g33u3
8<: ) g11g
11 ¼ 1
g22g
22 ¼ 1
g33g
33 ¼ 1
8<:
The linear operators gij, g
ij, gji are useful in the explanation of the more general
properties of tensors.
Exercise 1.13 For the vector v ¼ 4g1 þ 3g2 referenced to a coordinate system,
calculate their contravariant and covariant components in the referential system that
have the basis vectors g1 ¼ 3g1, g2 ¼ 6g1 þ 8g2, and g3 ¼ g3.
The metric tensor of the space is given by
gij ¼ gi � gj ¼
g1g1 g1g2 g1g3
g2g1 g2g2 g2g3
g3g1 g3g2 g3g3
24 35 ¼ 9 18 018 100 0
0 0 1
24 35) detgij ¼ 576
For the conjugated metric tensor
gijg
ij ¼ δij ) gij ¼ gij
h i�1
) gij ¼ 1
576
100 �180
�18 9 0
0 0 576
24 35
36 1 Review of Fundamental Topics About Tensors
The vectors of the contravariant basis are given by
gi ¼ gijgj
gi ¼
g1
g2
g3
8<:
9=; ¼ 1576
100 �18 0
�18 9 0
0 0 576
24 35 3g16g1 þ 8g2
g3
8<:
9=; ¼
1
3
g1 �
1
4
g2
1
8
g2
g3
8>>><>>>:
9>>>=>>>;
With the contravariant components of v
vi ¼ v � gi
v1 ¼ 4g1 þ 3g2ð Þ � g1 ¼
7
12
v2 ¼ 4g1 þ 3g2ð Þ � g2 ¼
3
8
v3 ¼ 4g1 þ 3g2ð Þ � g3 ¼ 0
so
v ¼ 7
12
g1 þ 3
8
g2
With the covariant components of v
vi ¼ v � gi ¼ 4g1 þ 3g2ð Þ � gi
v1 ¼ 4g1 þ 3g2ð Þ:g1 ¼ 12 v2 ¼ 4g1 þ 3g2ð Þ:g2 ¼ 48 v3 ¼ 4g1 þ 3g2ð Þ � g3 ¼ 0
so
v ¼ 12g1 þ 48g2
Exercise 1.14 Show that in the space EN exists g‘jgik � g‘igjk
� �
g‘j ¼ N � 1ð Þgik,
where gij is the metric tensor.
Developing the given expression
g‘jgik � g‘igjk
� �
g‘j ¼ g‘jgikg‘j � g‘igjkg‘j ¼ g‘jg‘jgik � g‘ig‘jgjk
¼ g‘jg‘jgik � g‘iδ ‘k ¼ g‘jg‘jgik � gki ¼ δ jj gik � gki
as δ jj ¼ δ11 þ δ22 þ � � � þ δ nn ¼ N for the space EN, the result is
g‘jgik � g‘igjk
� �
g‘j ¼ Ngik � gki
1.5 Metric Tensor 37
but gik ¼ gki; thus,
g‘jgik � g‘igjk
� �
g‘j ¼ N � 1ð Þgik Q:E:D:
Exercise 1.15 Show that in space EN exists g
ij ∂gij
∂xk þ gij ∂g
ij
∂xk ¼ 0, where gij is the
metric tensor.
The relation between the metric tensor and its conjugated tensor is given by
gijg
ij ¼ δ jj ¼ δ11 þ δ22 þ � � � þ δnn ¼ N
Differentiating this expression with respect to xk
∂ gijg
ij
� �
∂xk
¼ ∂gij
∂xk
gij þ gij
∂ gijð Þ
∂xk
¼ ∂N
∂xk
¼ 0
so
gij
∂gij
∂xk
þ gij
∂gij
∂xk
¼ 0 Q:E:D:
Exercise 1.16 For the symmetric tensor Tij, that fulfills the condition gijT‘k�
gi‘Tjk þ gjkT‘i � gk‘Tij ¼ 0, show that Tij ¼ mgij, where m 6¼ 0 is a scalar.
Multiplying the expression given by gij follows
gijgijT‘k � gijgi‘Tjk þ gijgjkT‘i � gijgk‘Tij ¼ δ ii T‘k � δ ‘j Tjk þ δ ikT‘i � gijgk‘Tij ¼ 0
As δ jj ¼ δ11 þ δ22 þ � � � þ δnn ¼ N, and for j ¼ ‘ and i ¼ k
NTji ¼ gijgijTij
As ds2 ¼ gijTij ¼ m1, where m1 is a scalar, and with Tij ¼ Tji follows
NTji ¼ m1gij ) Tji ¼
m1
N
gij
Putting m ¼ m1N
Tij ¼ mgij Q:E:D:
38 1 Review of Fundamental Topics About Tensors
1.6 Angle Between Curves
The angle between two curves is defined by the angle formed by their tangent unit
vectors g1, g2 (Fig. 1.9), by means of the dot product
cos α ¼ g1 � g2
g1k k g2k k
¼ g1 � g2
In differential terms this angle is calculated supposing that in the space E3 two
curves intersect in a point R, and admitting a third curve that intersects the other two
at points A1 and A2, which distances from the point R are, ds(1) and ds(2) (Fig. 1.9).
The points M,A1,A2 have coordinates x
i, xi þ dx i
1ð Þ and x
i þ dx i
2ð Þ, respectively.
With the cosine law
cos α ¼ lim RA1ð Þ
2 þ RA2ð Þ2 � A1A2ð Þ2
2 RA1ð Þ RA2ð Þ
which in differential terms stays
cos α ¼ ds 1ð Þ
� �2 þ ds 2ð Þ� �2 � ds 3ð Þ� �2
2ds 1ð Þds 2ð Þ
and using the basic expressions for the length of the arcs of the curves
ds 1ð Þ
� �2 ¼ gijdx i1ð Þdxj1ð Þ ds 2ð Þ� �2 ¼ gijdx i2ð Þdxj2ð Þ
ds 3ð Þ
� �2 ¼ gij xi þ dxi1ð Þ� �� xi þ dx i2ð Þ� �h i2 ¼ gij dx i1ð Þ � dxi2ð Þ� �2
¼ gij dx i2ð Þ � dx i1ð Þ
� �
dx j
2ð Þ � dxj1ð Þ
� �
1T 2
T
1C
2C
R
( )1ds
( )2ds
1A
2A
3C
( )3ds
Fig. 1.9 Angle between
two curves
1.6 Angle Between Curves 39
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then
cos α ¼
gij dx
i
1ð Þdx
j
2ð Þ þ dxj2ð Þdxi1ð Þ
� �
2ds 1ð Þds 2ð Þ
¼
gijdx
i
1ð Þdx
j
2ð Þ
ds 1ð Þds 2ð Þ
Considering ui ¼ dx
i
1ð Þ
ds 1ð Þ
and vj ¼ dx
j
2ð Þ
ds 2ð Þ
, which are, respectively, the contravariant
unit vectors of the tangents T1 and T2 to the curves C1 and C2, respectively, provides
cos α ¼ gij
dx i
1ð Þ
ds 1ð Þ
 !
dxj
2ð Þ
ds 2ð Þ
 !
If two vectors are orthogonal, then α ¼ π
2
, so the condition of orthogonality for
two directions is giju
ivj ¼ 0. The necessary and sufficient condition so that a
coordinate system is orthonormal is that gij ¼ 0 8i 6¼ j at the points of this space.
The null vector has the peculiar characteristic of being normal to itself.
Figure 1.10 illustrates the components of the differential element of arc ds with
respect to the coordinate system Xi with origin at point P. The lengths of the arc
elements measured with respect to the coordinate axes of the referential system are
ds 1ð Þ ¼ ffiffiffiffiffiffig11p dx1, ds 2ð Þ ¼ ffiffiffiffiffiffig22p dx2, and ds 3ð Þ ¼ ffiffiffiffiffiffig33p dx3.
To prove that α is real and that cos α 	 1, consider the expression
cos α ¼ gijuivj ¼ gijuivj ¼ uivj ¼ uivj
where u, v are unit vectors. Admit that these vectors are multiplied by two non-null
real numbers ‘,m, originating ‘ui þ mvið Þ, as the metric of the space is positive
definite, then for all the values of this pair of numbers
gij ‘u
i þ mvi� � ‘uj þ mvj� � � 0
1X
2X
3X
2
2 22 xdgds =
( )
1
111 xdgds =
( )
3
333 xdgds =
P
1 2a
13
a
2 3a
ds
Fig. 1.10 Components of
the differential arc element
with respect to the Xi
coordinate system
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Developing this inequality
‘2 þ 2‘m cos αþ m2 � 0
for uivj ¼ uivj ¼ cos α, which can be written under the form
‘þ m cos αð Þ2 þ m2 1� cos 2α� � � 0
that will be positive definite if cos 2α 	 1 or cos αk k 	 1, so α is real.
Let the modulus of a vector in terms of their contravariant components
v ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
εgk‘v
kv‘
q
ð1:6:1Þ
and in terms of their covariant components
v ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
εgkkvkvk
p
ð1:6:2Þ
thus the angle between two curves is determined when calculating the angle
between their tangent unit vector ui, vj, then
cos α ¼ giju
ivjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uiuið Þ vjvj
� �q ð1:6:3Þ
Exercise 1.17 Let the orthogonal unit vectors ui and vj, calculate the norm of
vector wi ¼ ui þ vi.
The condition of orthogonality between two vectors is given by
giju
ivj ¼ 0
and as ui and vj are unit vectors
giju
iuj ¼ 1 gijvivj ¼ 1
For the vector wi
wk k2 ¼ gijwiwj ¼ gij ui þ vjð Þ uj þ við Þ ¼ gijuiuj þ gijuivi þ gijvjuj þ gijvjvi
¼ 1þ 0þ 0þ 1 ¼ 2
then
wk k ¼
ffiffiffi
2
p
1.6 Angle Between Curves 41
Exercise 1.18 The vectors ui and vj are orthogonal, and each one of them has
modulus ‘, show that gpjgki � gpkgji
� �
upvjukvi ¼ ‘4.
The square of the modulus of the vectors is given by
giju
iuj ¼ ‘2 gijvivj ¼ ‘2
and the condition of orthogonality between these vectors is given by
giju
ivj ¼ 0
Developing the given expression
gpjgki � gpkgji
� �
upvjukvi ¼ gpjgkiupvjukvi � gpkgjiupvjukvi
¼ gpjupvj
� �
gkiu
kvi � gpkupuk
� �
gjiv
jvi
¼ 0� ‘2 � ‘2
so
gpjgki � gpkgji
� �
upvjukvi ¼ ‘4 Q:E:D:
Exercise 1.19 Given the symmetric tensor Tij and the unit vectors u
i and vj
orthogonal to the vector wk, show that Tiju
i � m1gijui þ n1gijwi ¼ 0 and Tijvi�
m2gijv
i þ n2gijwi ¼ 0, where m1 6¼ m2 and n1 6¼ n2 are scalars, then these unit
vectors are orthogonal.
As ui and vj are unit vectors, then
giju
iuj ¼ 1 gijvivj ¼ 1
and the conditions of orthogonality of these unit vectors with respect to the vector
wi are
giju
iwj ¼ 0 gijviwj ¼ 0
Multiplying by vj both the members of the first expression
Tiju
ivj � m1gijuivj þ n1gijwivj ¼ 0 ) Tijuivj ¼ m1gijuivj
and multiplying by uj both the members of the second expression
Tijv
iuj � m2gijviuj þ n2gijwiuj ¼ 0 ) Tijviuj ¼ m2gijviuj
42 1 Review of Fundamental Topics About Tensors
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The indexes i and j are dummies, so their position can be changed
Tiju
ivj ¼ m2gijuivj
thus
m1giju
ivj ¼ m2gijuivj ) m1 � m2ð Þgijuivj ¼ 0
As by hypothesis m1 6¼ m2, then gijuivj ¼ 0; this shows that the unit vectors ui
and vj are orthogonal.
1.6.1 Symmetrical and Antisymmetrical Tensors
If the change of position of two indexes, covariant or contravariant, does not modify
the tensor’s components, then this is a symmetrical tensor
T pqrsijk ¼ T pqrsikj ¼ T pqrsjik ¼ T pqrsjki ¼ T pqrskij ¼ T pqrskji .
The symmetry, a priori, does not ensure that the new variety is a tensor.Admit
that T pqrsijk‘ ¼ T qprsijk‘ , whereby by the hypothesis of this tensor’s symmetry, it follows
that T pqrsijk‘ � T qprsijk‘ ¼ 0. As Tpqrsijk‘ is a tensor, the result of the difference between the
two varieties being null, and as the referential system is arbitrary, it is concluded
that this result will always be null for any coordinate system, i.e., it always has the
tensor null. WritingT pqrsijk‘ þ 0 ¼ T qprsijk‘ , and as the summation of tensors is a tensor, it
is concluded that Tpqrsijk‘ is a tensor.
A tensor is called antisymmetrical with respect to two of its indexes, if it changes
signs on the change of position between these two indexes: Tijk‘ ¼ �T‘jki. The
number of independent components of an antisymmetric tensor of order p in the
space EN is given by
n ¼ N!
p! N � pð Þ ! ð1:7:1Þ
Let the space EN in which the antisymmetric pseudotensor of the third order ε
ijk
is defined (a general definition of pseudotensors will be presented in item 1.8), and
by the definition of antisymmetry, it provides six components of εijk which are
numerically equal:
εijk ¼ εjki ¼ εkij ¼ �εikj ¼ �εjik ¼ �εkji
This variety has 27 components, having 21 null, for it is verified that only the six
components ε123 ¼ ε231 ¼ ε312 ¼ �ε132 ¼ �ε213 ¼ �ε321 are non-null. Let, for
example, a linear and homogeneous transformation be applied to the component ε123:
1.6 Angle Between Curves 43
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ε123 ¼ ∂x
1
∂xi
∂x2
∂xj
∂x3
∂xk
ε123 ð1:7:2Þ
Developing expression (1.7.2)
ε123 ¼ ∂x
1
∂x1
∂x2
∂x2
∂x3
∂x3
þ ∂x
1
∂x2
∂x2
∂x3
∂x3
∂x1
þ ∂x
1
∂x3
∂x2
∂x1
∂x3
∂x2
�
�∂x
1
∂x1
∂x2
∂x3
∂x3
∂x2
� ∂x
1
∂x3
∂x2
∂x2
∂x3
∂x1
� ∂x
1
∂x2
∂x2
∂x1
∂x3
∂x3
�
ε123
ð1:7:3Þ
In compact form for the component ε123 in the coordinate system X
i
ε123 ¼ ∂x
k
∂x‘
				 				ε123 ð1:7:4Þ
and with
gk‘ ¼
∂xi
∂xk
∂xj
∂x‘
gij
It follows by means of product of determinants
gk‘j j ¼
∂xi
∂xk
				 				 � ∂xj∂x‘
				 				 � gij			 			) gk‘j j ¼ ∂xi∂xk
				 				2 gij			 			) g ¼ ∂xi∂xk
				 				2g
1ffiffiffi
g
p ¼ ∂x
i
∂xk
				 				 1ffiffiffigp ð1:7:5Þ
Comparing expression (1.7.5) with the expression (1.7.4)
ε123 ¼ 1ffiffiffi
g
p ð1:7:6Þ
So as to generalize expression (1.7.6), this analysis is made for the other
contravariant components of the pseudotensor εijk. As this variety assumes the
values 0, 
 1 as a function of the position of their indexes, it is linked to the
permutation symbol eijk by means of the following relations:
εijk ¼
þ1ffiffiffi
g
p eijk is an even permutation of the indexes
�1ffiffiffi
g
p eijk is an odd permutation of the indexes
0 when there are repeated indexes
8>>>><>>>>: ð1:7:7Þ
44 1 Review of Fundamental Topics About Tensors
Expression (1.7.7) represents the components of the Ricci pseudotensor, also
called Levi-Civita pseudotensor. The covariant components of this pseudotensor
are obtained by means of the metric tensor, whereby using the approaches presented
in item 1.5, it is provided for the lowering of the indexes of the pseudotensor εpqr:
εijk ¼ gipgjqgkrεpqr ð1:7:8Þ
and with the definition of the determinant of the metric tensor, and with the
definition of εijk given by the relations (1.7.7), it follows that
εijk ¼ gij
			 			 1ffiffiffi
g
p ¼ ffiffiffigp ð1:7:9Þ
In terms of the permutation symbol eijk, it is provided as the covariant coordi-
nates of the Ricci pseudotensor
εijk ¼
ffiffiffi
g
p
eijk is an even permutation of the indexes
� ffiffiffigp eijk is an odd permutation of the indexes
0 when there are repeated indexes
8<: ð1:7:10Þ
The definition of the Ricci pseudotensor presented for the space E3 is general-
ized for the space EN, in which the contravariant components and covariant of this
variety are given, respectively, in terms of the permutation symbol by
εi1i2i3���in ¼
þ1ffiffiffi
g
p ei1i2i3���in is an even permutation of the indexes
�1ffiffiffi
g
p ei1i2i3���in is an odd permutation of the indexes
0 when there are repeated indexes
8>>>><>>>>: ð1:7:11Þ
εi1i2i3���in ¼
ffiffiffi
g
p
ei1i2i3���in is an even permutation of the indexes
� ffiffiffigp ei1i2i3���in is an odd permutation of the indexes
0 when there are repeated indexes
8<: ð1:7:12Þ
The conception of permutation symbol is associated to the value of a determi-
nant, with no link to the space EN, whereby it refers only to a symbol that seeks to
simplify the calculations. With the definition of the Ricci pseudotensor in terms of
this symbol, it is verified that in the relation between these two varieties exists the
term
ffiffiffi
g
p
linked to the metric of the space. This shows the fundamental difference
between the same, for the change of sign of the Ricci pseudotensor as a function of
the permutations of their indexes (sign defined by the permutation symbol) indi-
cates the orientation of the space.
With relation (1.7.10) it follows that
εijkεjki ¼ 3! ¼ 6 ð1:7:13Þ
1.6 Angle Between Curves 45
The definitions and deductions presented next seek to complement the relations
between the generalized Kronecker delta and the Ricci pseudotensor in the space
EN. These expressions are called δ� ε relations.
1.6.1.1 Generalization of the Kronecker Delta
The Ricci pseudotensor represents the mixed product of three vectors ∂x
i
∂x‘ ,
∂xj
∂x‘ ,
∂xk
∂x‘,
where ‘ ¼ 1, 2, 3 indicates the components of these vectors, which comprise the
lines and columns of the determinant that expresses this product, called Gram
determinant, that in terms of their covariant components stays
εijk ¼ ∂x
i
∂x‘
� ∂x
j
∂x‘
� �
� ∂x
k
∂x‘
¼
∂xi
∂x1
∂xj
∂x1
∂xk
∂x1
∂xi
∂x2
∂xj
∂x2
∂xk
∂x2
∂xi
∂x3
∂xj
∂x3
∂xk
∂x3
											
											
¼
δ1i δ1j δ1k
δ2i δ2j δ2k
δ3i δ3j δ3k
						
						
and in terms of their contravariant components
εpqr ¼
∂xp
∂x1
∂xq
∂x1
∂xr
∂x1
∂xp
∂x2
∂xq
∂x2
∂xr
∂x2
∂xp
∂x3
∂xq
∂x3
∂xr
∂x3
											
											
¼
δ1p δ1q δ1r
δ2p δ2q δ2r
δ3p δ3q δ3r
						
						
The product of these two determinants being given by
εijkε
pqr ¼
δ1i δ1j δ1k
δ2i δ2j δ2k
δ3i δ3j δ3k
						
						 �
δ1p δ1q δ1r
δ2p δ2q δ2r
δ3p δ3q δ3r
						
						
εijkε
pqr ¼
δ1iδ
1p þ δ2iδ2p þ δ3iδ3p
� �
δ1iδ
1q þ δ2iδ2q þ δ3iδ3q
� �
δ1iδ
1r þ δ2iδ2r þ δ3iδ3r
� �
δ1jδ
1p þ δ2jδ2p þ δ3jδ3p
� �
δ1jδ
1q þ δ2jδ2q þ δ3jδ3q
� �
δ1jδ
1r þ δ2jδ2r þ δ3jδ3r
� �
δ1kδ
1p þ δ2kδ2p þ δ3kδ3p
� �
δ1kδ
1q þ δ2kδ2q þ δ3kδ3q
� �
δ1kδ
1r þ δ2kδ2r þ δ3kδ3r
� �
						
						
εijkε
pqr ¼
δmiδ
mp δmiδ
mq δmiδ
mr
δmjδ
mp δmjδ
mq δmjδ
mr
δmkδ
mp δmkδ
mq δmkδ
mr
						
						
εijkε
pqr ¼
δpi δ
q
i δ
r
i
δpj δ
q
j δ
r
j
δpk δ
q
k δ
r
k
						
						 ð1:7:14Þ
46 1 Review of Fundamental Topics About Tensors
With the expressions (1.7.10) and (1.7.7), it follows that
εr‘mε
rst ¼ δrstr‘m ¼ δst‘m ¼
1
0
�1
8<: ð1:7:15Þ
The contraction of the indexes k and r of the product of two pseudotensors, given
by expression (1.7.15), provides
εijkε
pqr ¼ δpqkijk ¼
δ pi δ
q
i δ
k
i
δ pj δ
q
j δ
k
j
δ pk δ
q
k δ
k
k
						
						
and as δ kk ¼ 3 it follows that
εijkε
pqr ¼ δpqkijk ¼
δ pi δ
q
i δ
k
i
δ pj δ
q
j δ
k
j
δ pk δ
q
k 3
						
						 ¼ δ
p
i δ
q
i
δpj δ
q
j
				 				 ¼ δpi δqj � δ qi δpj ð1:7:16Þ
Analogously, and with the contraction of the indexes j and p:
εijkε
pqr ¼ δjqrijk ¼
δ ji δ
q
i δ
k
i
3 δqj δ
k
j
δ jk δ
q
k δ
k
k
						
						 ¼ � δ
q
i δ
r
i
δqk δ
r
k
				 				 ¼ δ ri δqk � δqi δ rk
The product εijrεpqr ¼ δpqij leads to the generalization of the Kronecker delta that
has its value defined as a function of the number of permutations of their indexes.
For the covariant components of this operator, δijpq ¼ δpijq is provided, where the
number of permutations of the indexes is even, so it is verified that this operator is
symmetrical, and δijpq ¼ �δjipq ¼ �δijqp is antisymmetric for an odd number of
permutations of the indexes. The deltas with repeated indexes are null, for example,
δ11pq ¼ δ22pq ¼ δij33 ¼ 0. This analysis allows defining the generalized Kronecker
delta inspace EN:
δ
j
1
j2j3���jn
i1i2i3���in ¼
þ1 is an even permutation of i1i2i3� � �, j1j2j3� � �
�1 is an odd permutation of i1i2i3� � �, j1j2j3� � �
0 when there are repeated indexes
8<: ð1:7:17Þ
1.6.1.2 Fundamental Expressions with the Generalized
Kronecker Delta
The generalized Kronecker delta in terms of the Ricci pseudotensor is given by
1.6 Angle Between Curves 47
εi1i2i3���imε
j
1
j2j3���jm ¼ δj1 ���jmjmþ1���jni1���imimþ1���in ¼
δj1i1 δj1i2 � � � δj1in
δj2i1 δj2i2 � � � δj2in
� � � � � � � � � � � �
δjni1 δjni2 δj1i1 δjnin
								
								 ð1:7:18Þ
Various fundamental expressions are obtained with the Kronecker delta δpqij that
are useful in Tensor Calculus. Let, for example, the contraction of the indexes j and
q of this tensor
δpqij ¼ δpjij ¼ δpi δ jj � δ qi δ jj ¼ δpi δ jj � δ ji δpj ¼ 3δpi � δ ji δ pj ¼ 2δ pi
whereby
δpi ¼
1
2
δpjij ¼
1
2
δp1i1 þ δp2i2 þ δp3i3
� �
It is also verified for the contractions j ¼ q and k ¼ r
δpqrijk ¼
1
2
δpjkijk ¼ δpi ¼
1
2
δp12i12 þ δp23i23 þ δp31i31
� �
The generalization of these expressions that involve Kronecker deltas for the
space EN is given by the following expression:
δ
p
1
p2p3���pm
i1i2i3���im ¼
N � nð Þ !
N � mð Þ ! δ
p
1
���pmpmþ1���pn
i1���imimþ1���in ð1:7:19Þ
The Kronecker delta tensor δ
p
1
p2p3���pm
i1i2i3���im provided by expression (1.7.19) is of order
2 n� mð Þ inferior to the order Kronecker delta tensor δp1 ���pmpmþ1���pni1���imimþ1���in , from which it
was obtained by means of contractions of the indexes. In this expression for
m ¼ 1, n ¼ 3
δpi ¼
1
N � 2ð Þ N � 1ð Þ δ
pjk
ijk
Putting m ¼ 1, n ¼ 2 in expression (1.7.19) results in
δ pi ¼
1
N � 1ð Þ δ
pj
ij
These two examples show that δpi can be obtained by two contractions of the
indexes of the sixth-order tensor δpqrijk or by means of only a contraction of
the indexes of the fourth-order tensor δpqij .
48 1 Review of Fundamental Topics About Tensors
In expression (1.7.19) for m ¼ 1, i ¼ p
δi1i1 ¼
N � nð Þ !
N � 1ð Þ ! δ
i1i2���in
i1i2���in
and as δi1i1 ¼ n it results in
n ¼ N � nð Þ !
N � 1ð Þ ! δ
i1i2���in
i1i2���in ) δi1i2���ini1i2���in ¼
n N � 1ð Þ !
N � nð Þ !
δi1i2���ini1i2���in ¼
n !
N � nð Þ ! ð1:7:20Þ
For the inner product of the Ricci pseudotensors εi1i2���in and ε
i1i2���in with N ¼ n
expression (1.7.20) provides
εi1i2���in ε
i1i2���in ¼ δi1i2���ini1i2���in ¼ n! ð1:7:21Þ
Expression (1.7.19) with N ¼ n provides
εi1i2���imimþ1���in ε
p1p2���pmpmþ1���pn ¼ N � mð Þ !δp1p2���pni1i2���in ð1:7:22Þ
Expression (1.7.22) relates in space EN the inner product of two Ricci
pseudotensors with the generalized Kronecker delta tensor.
1.6.1.3 Product of the Ricci Pseudotensor by the Generalized
Kronecker Delta
The definition of the generalized Kronecker delta shows that δ pijkq123 ¼ 0, for a
dummy index will always occur when these vary. With expression (1.7.18)
εq123ε
pijk ¼ δ pijkq123 ¼
δpq δp1 δp2 δp3
δiq δi1 δi2 δi3
δjq δj1 δj2 δj3
δkq δk1 δk2 δk3
								
								 ¼ 0
Developing this determinant in terms of the first column
δpqεijkε123 � δiqεpjkε123 þ δjqεpikε123 � δkqεpijε123 ¼ 0
and with εpik ¼ �εipk
εijkδpq ¼ εpjkδiq þ εipkδjq þ εijpδkq ð1:7:23Þ
1.6 Angle Between Curves 49
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The symmetry of δpq allows changing the position of these indexes
εijkδqp ¼ εqjkδip þ εiqkδjp þ εijqδkp ð1:7:24Þ
1.6.1.4 Norm of the Antisymmetric Pseudotensor of the Second Order
A vector is represented by an oriented segment of a straight line, and its norm is
given by the length of this segment. For an antisymmetric pseudotensor of the
second order A associated to an axial vector u provides that its norm is linked to the
area of the parallelogram which sides are the vectors that define the vectorial
product u ¼ v� w.
Let α the angle between the vectors v and w, the square of the modulus of the
cross product of these vectors is given by
uk k2 ¼ v� wk k2 ¼ vk k2 wk k2 sin 2α ¼ vk k2 wk k2 1� cos 2α� �
¼ vk k2 wk k2 � v � wð Þ2
thus
uk k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vk k2 wk k2 � v � wð Þ2
q
ð1:7:25Þ
This norm can be expressed in terms of the components of the pseudotensor A.
Let the components of the vectors v and w in the coordinate system Xk, so with the
expression (1.7.25)
uk k2 ¼ gi‘viv‘
� �
gjmw
jwm
� �
� gimviwmð Þ gj‘vjw‘
� �
¼ gi‘gjm � gimgj‘
� �
viv‘wjwm
¼
gi‘ gim
gj‘ gjm
					
					viv‘wjwm
This determinant allows writing
gi‘ gim
gj‘ gjm
				 				viv‘wjwm ¼ 12 gi‘ gimgj‘ gjm
				 				 viwj � vjwi� �v‘wm
and as
A‘m ¼ 1
2
v‘wm � vmw‘� �
it follows that
50 1 Review of Fundamental Topics About Tensors
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gi‘ gim
gj‘ gjm
				 				Aijv‘wm ¼ 12 gi‘ gimgj‘ gjm
				 				Aij v‘wm � vmw‘� �
then
gi‘ gim
gj‘ gjm
				 				viw‘v‘wm ¼ 12 gi‘ gimgj‘ gjm
				 				AijA‘m ð1:7:26Þ
1.6.1.5 Generation of Tensors from the Ricci Pseudotensor
The Ricci pseudotensor generates an antisymmetric tensor from a pseudotensor
(axial vector), and this pseudotensor generates an antisymmetric tensor from a
pseudotensor (axial vector). This characteristic of the Ricci pseudotensor in space
E3 is generalized for the space EN, where the known antisymmetric tensor A i1i2���in½ �
provides
Tj1j2���jn�m ¼ 1
m!
εj1j2���jn�mi1i2���imA i1i2���im½ � ð1:7:27Þ
Tensor Tj1j2���jn�m is generated by the Ricci pseudotensor, which works as an
operator applied to the antisymmetric tensor to produce this associated tensor.
Multiplying both the members of expression (1.7.27) by εj1j2���jn�mi1i2���im results in
εj1j2���jn�mi1i2���imT
j1j2���jn�m ¼ 1
m!
εj1j2���jn�mi1i2���imε
j1j2���jn�mi1i2���imA i1i2���im½ �
With expressions (1.7.19), (1.7.21), and (1.7.22), it follows that the expression
for the antisymmetric tensor A i1i2���im½ � in terms of the Ricci pseudotensor is given by
A i1i2���im½ � ¼
1
n� mð Þ ! εj1j2���jn�mi1i2���imT
j1j2���jn�m ð1:7:28Þ
To illustrate the application of expression (1.7.28), let the antisymmetric tensor
of the fourth order A[ijk‘] with i, j, k, ‘ ¼ 1, 2, 3, � � �n, to which the following five
varieties are associated
T ¼ 1
4!
εijk‘A ijk‘½ � Ti ¼ 1
4!
εijk‘pA jk‘p½ � Tij ¼ 1
4!
εijk‘pqA k‘pq½ �
Tijk ¼ 1
4!
εijk‘pqrA ‘pqr½ � Tijk‘ ¼ 1
4!
εijk‘pqrsA pqrs½ �
1.6 Angle Between Curves 51
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1.7 Relative Tensors
The tensors defined in the previous items are called absolute tensors. However,
other varieties with properties that are analogous to those of these tensors can be
defined. The relative tensors are more general varieties, the absolute tensors being a
particular case of the same.
In solving various problems that involve integration processes, the need of
generalizing the concept of tensor is verified. This generalization leads to the
concept of relative tensor. To exemplify the concept of the relative tensor, let a
covariant tensor of the second order be defined in the space E3, which transforms by
means of the expression
Tk‘ ¼ ∂x
i
∂xk
∂xj
∂x‘
Tij ð1:8:1Þ
The determinants of the terms of this function are given by detTk‘, det
∂xi
∂xk
� �
,
det ∂x
j
∂x‘
� �
, and det Tij. Applying the determinant product rule to the determinant
terms of this expression
detTk‘ ¼ det ∂x
i
∂xk
� �
det
∂xj
∂x‘
� �
detTij ð1:8:2Þ
the Jacobian of the inverse transformation of tensor Tk‘ is given by
J ¼ det ∂x
m
∂xn
� �
> 0 ð1:8:3Þ
so
detTk‘ ¼ J2detTij ð1:8:4Þ
Expression (1.8.4) shows that detTk‘ of a second-order tensor is not a scalar and
also is not a second-order tensor of the type Tpq. This expression is the new
transformation. Assuming that detTij > 0 and detTk‘ > 0 provides
detTk‘
� �1
2 ¼ J detTij
� �1
2 ð1:8:5Þ
This shows that the definition of tensors can be expanded introducing the concept of
relative tensor.
Consider the mixed tensor
T
ij...p
rs...v ¼ Jð ÞW
∂xi
∂xa
∂xj
∂xb
� � �∂x
p
∂xd
∂xr
∂xe
∂xs
∂xf
� � �∂x
v
∂xh
Tab...def ...h ð1:8:6Þ
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that is called relative tensor of weight W or with weighing factor W. This weight is
an integer number, and J is the Jacobian of the transformation. For the particular
case in which W ¼ 0, an absolute tensor exists.
The concept of relative tensor allows distinguishing a relative invariant of a
scalar, which is an absolute invariant. To differentiate these concepts, let the
relative invariant A of weight W, which transforms according to the expression
A ¼ JWA ð1:8:7Þ
For the particular case in whichW ¼ 0, an absolute tensor exists A ¼ A that is a
scalar. For W ¼ 1 provides the scalar density A ¼ JA. The definition of scalar
density will be presented in detail in later paragraphs.
To illustrate the concept of relative tensor, let the metric tensor gij with
detgij ¼ g. Applying a linear and homogeneous transformation to this tensor
eg‘m ¼ ∂xi∂ex‘ ∂xj∂exm gij ð1:8:8Þ
with detegij ¼ eg, and by means of the property of the product of determinants,
provides the relative scalar of weight W ¼ 2
eg ¼ J2g ) ffiffiffiegp ¼ J ffiffiffigp ð1:8:9Þ
For J ¼ 1 provides ffiffiffigp that is a relative tensor of unit weight, being, therefore, an
invariant. With expression (1.8.9) and the condition gg ¼ 1, having detgij ¼ g, it is
verified that
ffiffiffi
g
p
is a relative tensor of weight �1.
Let the Jacobian J of weightW ¼ 1, which is an invariant and when changing to
a new coordinate system provides for this determinant J ¼ αJ being α a scalar
(invariant). Raising both members of this expression to the power W
J
W ¼ αWJW ð1:8:10Þ
where JW is an invariant of weight W, thus
αW ¼ JWJ�W ð1:8:11Þ
Consider the relative tensor Tijk of weight W that transforms by means of the
expression
T
i
jk ¼ αW
∂xm
∂xj
∂xn
∂xk
∂xi
∂x‘
T ‘mn ð1:8:12Þ
and substituting in expression (1.8.12), the value of αW given by expression (1.8.11)
provides
1.7 Relative Tensors 53
T
i
jk ¼ J
W
J�W
� �∂xm
∂xj
∂xn
∂xk
∂xi
∂x‘
T ‘mn
It follows that
J
�W
T
i
jk
� �
¼ ∂x
m
∂xj
∂xn
∂xk
∂xi
∂x‘
J�WT ‘mn
� � ð1:8:13Þ
As J�WT ‘mn
� �
is an absolute tensor, and by means of the transformation law of
tensors, it is concluded that J
�W
T
i
jk is also an absolute tensor. This shows that the
transformation of a relative tensor of weight W in an absolute tensor is carried out
multiplying it by the invariant of unit weight raised to the power�W. The invariantffiffiffi
g
p
of unit weight is used to carry out this kind of transformation. This systematic
allows, for instance, transforming the relative tensor Tkij of weight W into the
absolute tensor Akij by means of
T kij
ffiffiffi
g
p� ��W ¼ Akij ð1:8:14Þ
The operations multiplying by a scalar, addition, subtraction, contraction, outer
product, and inner product are applicable to the relative tensors. These operations
provide new relative tensors; as a result, the proof is analogous to the demonstra-
tions performed for the absolute tensors.
Exercise 1.20 Show that δij is an absolute tensor.
It is admitted firstly that δij is a relative tensor of unit weight, being detδij ¼ 1,
which transforms into the absolute tensor δ∗ij by means of the expression
δ∗ij ¼
ffiffiffi
g
p� ��1
δij
Asδ∗ij is an isotropic tensor so δij is also isotropic, then
ffiffiffi
g
p� ��1 ¼ 1, which shows
that δij is an absolute tensor.
1.7.1 Multiplication by a Scalar
This operation provides as a result a relative tensor of weightW, which components
are the components of the original relative tensor multiplied by the scalar.
Let, for example, the relative tensor (J )WTij and the scalar m which product Pij is
given by Jð ÞWPij ¼ m Jð ÞWTij. To demonstrate that this expression represents a
tensor, it is enough to apply the transformation law of tensors to this expression.
54 1 Review of Fundamental Topics About Tensors
1.7.1.1 Addition and Subtraction
This operation is defined for relative tensors of the same order and of the same kind,
such as in the case of the following mixed tensors
Jð ÞWT kij ¼ Jð ÞWAkij þ Jð ÞWBkij
Subtraction is defined in the same way as addition, however, admitting that one
of the tensors be multiplied by the scalar�1: Jð ÞWT kij ¼ Jð ÞWAkij þ �1ð Þ Jð ÞWBkij . To
demonstrate that these expressions represent relative tensors, the transformation
law of tensors is applied to this expression.
1.7.1.2 Outer Product
This operation is defined in the same way as the outer product of absolute tensors.
Let, for example, the relative tensor Jð ÞW1Ak...ij... of variance ( p, q) and weightW1, and
the relative tensor Jð ÞW2B...‘m...rs of variance (u, v) and weight W2, which multiplied
provide
Jð ÞWTk...‘mij...rs ¼ Jð ÞW1Ak...ij...
h i
Jð ÞW2B...‘m...rs
h i
that is a relative tensor of variance pþ u, qþ vð Þ and weight W ¼ W1 þW2. To
demonstrate that this product is a relative tensor, the transformation law of tensors
is applied to this expression.
1.7.1.3 Contraction
This operation is defined in the same way as the contraction of the absolute tensors.
Let, for example, the relative tensor (J)WTijk‘m, in which contracting the upper index
j provides
Jð ÞWTijj‘m ¼ Jð ÞWT i‘m
that shows that the resulting relative tensor has its order reduced in two, but
maintains its weight W. To demonstrate that this contraction is a relative tensor,
the transformation law of tensors is applied to this expression.
1.7 Relative Tensors 55
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1.7.1.4 Inner Product
This operation is defined in the same manner as the inner product of the absolute
tensors.
Let, for example, two relative tensors Jð ÞW1Aij and Jð ÞW2B ‘k , whereby it follows
for the outer product of these tensors
Jð ÞW1þW2P ‘ijk ¼ Jð ÞW1Aij
h i
Jð ÞW2B ‘k
h i
that represents a relative tensor of the fourth order and weight W ¼ W1 þW2, and
with the contraction of the index ‘ the inner product is given by
Jð ÞW1þW2P ‘ij‘ ¼ Jð ÞW1Aij
h i
Jð ÞW2B ‘‘
h i
¼ Jð ÞW1þW2Pij
This shows that the resulting relative tensor is of the second order and weight
W ¼ W1 þW2. To demonstrate that this product is a relative tensor, the transfor-
mation law of tensors is applied to this expression.
1.7.1.5 Pseudotensor
The varieties that present a few tensorial characteristics, for example, when chang-
ing the coordinate system they follow a transformation law that differs from the
transformation law of tensors by the presence of the Jacobian, are called
pseudotensor (relative tensors). However, these varieties are not maintained invari-
ant when the coordinate system is transformed.
The definitions of the antisymmetric pseudotensors εijk and ε
ijk are associated,
respectively, to the permutation symbols in the covariant form eijk or in the
contravariant form eijk, to which correspond the values þ1 or �1 relative to the
even or odd number of permutations, respectively. The Ricci pseudotensors εijk and
εijk are associated to the concept of space orientation.
These varieties, when changing the coordinate system, transform in the same
way as the tensors, but are not invariant after these transformations. This shows that
a few characteristics are similar to the tensors but vary with the change of referen-
tial, for they assume the values 
1, so they are not tensors in the sensu stricto of
the term.
In expression (1.7.27) it is verified that εj1j2���jn�mi1i2���im has weight þ1, and the
tensor Tj1j2���jn�m has weight superior to the weight of the antisymmetric tensor
A i1i2���im½ �. This expression illustrates the applying of the pseudotensors.
Exercise 1.21 Show that
(a) εijk is a covariant pseudotensor of the third order and weight �1.
(b) εijk is a contravariant pseudotensor of the third order and weight þ1.
(c) The absolute pseudotensors can be obtained from these pseudotensors.
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(a) The definition of determinant allows writing Jεpqr, and as the pseudotensor εijk
assume the values 0,
1, on being applied to this variety,it provides a linear and
homogeneous transformation
Jεpqr ¼ ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
εijk ) εpqr ¼ ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
J�1εijk ¼ εpqr
then εijk is a covariant pseudotensor of the third order and weight �1.
(b) In a way that is analogous to the previous case, for defining the determinant
Jε‘mn, and for the transformation law of tensors
Jεpqr ¼ ∂x
p
∂xi
∂xq
∂xj
∂xr
∂xk
εijk ) εpqr ¼ ∂x
p
∂xi
∂xq
∂xj
∂xr
∂xk
J
�1
εijk
As JJ ¼ 1 it results in
εpqr ¼ ∂x
p
∂xi
∂xq
∂xj
∂xr
∂xk
Jεijk
then εijk is a contravariant pseudotensor of the third order and weight +1.
(c) As the pseudotensor εijk has weight �1, it follows by the transformation law of
relative tensors into absolute tensors, where the upper asterisk indicates the
absolute tensor
ε*ijk ¼
ffiffiffi
g
p� ��1h i�1
εijk ¼ ffiffiffigp εijk
For the relative pseudotensor εijk the absolute pseudotensor indicated by the
lower asterisk exists
εijk* ¼
1ffiffiffi
g
p εijk
Exercise 1.22 Show that gij is an absolute tensor.
Rewriting expression (1.8.9) ffiffiffi
g
p
¼ J ffiffiffigp
and with the cofactor of the matrix of tensor gij given by
Gij ¼ 1
2
eik‘ejpqgkpg‘q
and in terms of Ricci’s pseudotensor
1.7 Relative Tensors 57
Gij ¼ 1
2
εik‘εjpqgkpg‘q
it follows that
gij ¼ G
ij
g
¼ 1
2
εik‘εjpqgkpg‘q
The term to the right of this expression. is the product of two pseudotensors and
the tensors, being gij the inner product of these two varieties. This expression has
weight W ¼ 0, then gij is an absolute tensor.
1.7.1.6 Scalar Capacity
Let an antisymmetric pseudotensor Cijk in an affine space, for which according to
expression (1.7.1) for N ¼ 3 and p ¼ 3, there is only one independent component.
Writing Cijk as a function εijk follows
Cijk ¼ εijkc
where c is a component of the variety and with the change of the coordinate system
Cijk ¼ ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
C
pqr
and as the antisymmetry is maintained when the reference system is changed
C
pqr ¼ εpqrc ð1:8:15Þ
Considering the component C123:
C123 ¼ ∂x
1
∂xp
∂x2
∂xq
∂x3
∂xr
C
pqr ð1:8:16Þ
and substituting expression (1.8.15) in expression (1.8.16)
c ¼ εpqr ∂x
1
∂xp
∂x2
∂xq
∂x3
∂xr
c ð1:8:17Þ
Let
J ¼ εpqr ∂x
1
∂xp
∂x2
∂xq
∂x3
∂xr
ð1:8:18Þ
58 1 Review of Fundamental Topics About Tensors
results in the following expressions
c ¼ Jc ) c ¼ 1
J
c ¼ Jc ð1:8:19Þ
Function c is the only independent component of the antisymmetric
pseudotensor Cijk, which is called scalar capacity. Then a scalar capacity is a
pseudotensor of weight �1. To illustrate the concept of scalar capacity, let, for
example, the antisymmetric variety dVijk that defines an elementary volume in
space E3. This analysis follows the same routine presented when defining the scalar
capacity. The elementary volume is obtained by means of the mixed product of
three vectors that define the three reference axes in this space
dVijk ¼
dx1 0 0
0 dx2 0
0 0 dx3
						
						 ¼ dx1dx2dx3 ¼ dV ) dVijk ¼ dxidxjdxk ð1:8:20Þ
and with the transformation law of tensors
dVijk ¼ ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
dxpdxqdxr dV ¼ ∂x
1
∂xp
∂x2
∂xq
∂x3
∂xr
dxpdxqdxr ð1:8:21Þ
The antisymmetry of the pseudotensor is maintained when changing the coor-
dinate system
dxpdxqdxr ¼ εijkdV ð1:8:22Þ
and substituting expression (1.8.21) in expression (1.8.22)
dV ¼ εijk ∂x
1
∂xp
∂x2
∂xq
∂x3
∂xr
dV
results in the following expressions
dV ¼ JdV ) dV ¼ 1
J
dV ∴dx1dx2dx3 ¼ 1
J
dx1dx2dx3 ð1:8:23Þ
This shows that the elementary volume in an affine space is a pseudoscalar of
weight�1. In a more restricted manner, it says that the volume is a scalar capacity.
The term capacity comes from the association of the volume (capacity, content) to
the variety being analyzed. It is concluded that the integration of expression
(1.8.23), which represents a scalar field, is a pseudoscalar.
1.7 Relative Tensors 59
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1.7.1.7 Scalar Density
Let the antisymmetric pseudotensor Dijk, for which an analysis analogous to the one
developed when defining the scalar capacity is carried out
D123 ¼ D ¼ εijk ∂x
i
∂x1
∂xj
∂x2
∂xk
∂x3
D
J ¼ εpqr ∂x
1
∂xp
∂x2
∂xq
∂x3
∂xr
¼ εijk ∂x
i
∂x1
∂xj
∂x2
∂xk
∂x3
then
D ¼ JD ) D ¼ 1
J
D ) D ¼ JD
Function D is the unique component of the antisymmetric pseudotensor Dijk,
which is called scalar density. Then a scalar density is a pseudotensor of weightþ1.
To illustrate the concept of scalar density, let, for example, a body of elementary
mass dm in the affine space E3. This mass is determined by means of density
(specific mass) ρ(x1, x2, x3) and the elementary volume dV, thus
dm ¼ ρ x1; x2; x3ð ÞdV.
Considering that the mass is invariable (is a scalar)
dm ¼ ρ x1; x2; x3� �dV ¼ ρ x1; x2; x3� �dV ð1:8:24Þ
and as dV is a scalar capacity of weight �1, substituting expression (1.8.23) in
expression (1.8.24) provides
ρ x1; x2; x3
� �
dV ¼ ρ x1; x2; x3� � 1
J
dV ) ρ x1; x2; x3� � ¼ J ρ x1; x2; x3� � ð1:8:25Þ
This shows that the density in an affine space is a pseudoscalar of weightþ1, this
variety being called scalar density. The term density is not physically correct, for in
truth ρ(x1, x2, x3) such as it is presented defines the body’s specific mass.
The concepts shown for the elementary volume and for density in the affine
space E3 can be generalized for the space EN. The varieties that transform by means
of the expressions with structure analogous to the structures of expressions (1.8.23)
and (1.8.25) are called, respectively, scalar capacity and scalar density in space EN.
1.7.1.8 Tensorial Capacity
Let the space E3 where product of a scalar capacity c by the tensor T
k
ij exists, which
defines a tensorial density Ckij given by
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Ckij ¼ cT kij ð1:8:26Þ
and for a new coordinate system
C
r
pq ¼
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
Ckij ð1:8:27Þ
it follows that
C
r
pq ¼
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
cT kij ¼
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
JcT kij ) C
r
pq ¼ JcT rpq ð1:8:28Þ
Expression (1.8.28) shows that Ckij transforms in accordance with a law that is
similar to the transformation law of scalar capacity; however, it does not represent a
relative scalar but a relative tensor of weightþ1. The generalization of the concepts
of tensorial capacity for the space EN is immediate.
1.7.1.9 Tensorial Density
Let, for example, the space E3 where the product of a scalar densityD by the tensor
Tkij exists, which defines a tensorial density D
k
ij given by
Dkij ¼ DT kij ð1:8:29Þ
and for a new coordinate system
D
r
pq ¼
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
Dkij ¼
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
DT kij ¼
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
1
J
DT kij
D
r
pq ¼ J�1DT
r
pq ð1:8:30Þ
Expression (1.8.30) shows that Dkij transforms in accordance with a law that is to
the scalar density transformation law. However, it does not represent a relative
scalar but a relative tensor of weight �1. The generalization of the concepts of
tensorial density for the space EN is immediate.
The outer products between these varieties (pseudotensors and tensors) result in
Scalar capacity� scalar density ¼ scalar
Scalar capacity� tensor ¼ tensorial capacity
Scalar density� tensor ¼ tensorial density
Pseudotensor� pseudotensor ¼ tensor
Tensor� pseudotensor ¼ pseudotensor
1.7 Relative Tensors 61
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1.8 Physical Components of a Tensor
In mathematics the approach to the problems, in general, is carried out by means of
nondimensional parameters. In physics and engineering the parameters have mag-
nitude and dimensions, for example, N/mm2,m/s, etc. The analysis of a physical
problem by tensorial means requires that the parameters being studied be invariant
when changing the coordinate system. It happens that the axes of the coordinate
systems generally do not have the same dimensions. A Cartesian coordinate system
has axes that define lengths, but, for example, a spherical coordinate system has two
axes that express nondimensional coordinates, the same occurring in thecylindrical
coordinate system with one of their axes. Therefore, the components of a tensor
have dimensions, and when the coordinate system is changed, these components
vary in magnitude and dimension.
To express the transformation of tensors in a consistent way (in magnitude and
dimension), and that these varieties can be added after a change of the coordinate
systems, the same must be expressed in terms of their physical components.
1.8.1 Physical Components of a Vector
The concept of geometric vector is associated to the idea of displacement, its
transformation law being dxk ¼ ∂xk
∂xi
dxi where the coefficients ∂x
k
∂xi
are constants.
With respect to a Cartesian coordinates, the term dxjgj represents a displacement in
terms of the unit vectors of the coordinate axes
dxjgj ¼ dx1iþ dx2jþ dx3k ð1:9:1Þ
However, this term in a curvilinear coordinates does not represent a displace-
ment, so gk will not be a unit vector in this coordinate system. This shows that the
vector must be written in terms of components that express a displacement, called
the physical components of the vector.
Consider the vector u with physical components u�j , which can be written in
terms of these components and of their contravariant components
u ¼ u*kek ¼ ukgk ð1:9:2Þ
Comparing expressions (1.9.1) and (1.9.2)
dxjgj ¼ dx*jej ð1:9:3Þ
where dx* j are the physical components which by analogy correspond to the
displacement dxk, and with the unit vectors of base gk, ek, the components u
k, u�k
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are obtained in terms of the unit vector gk. Letgi � gj ¼ gij then gj
�� �� ¼ ffiffiffiffiffiffiffiffig jjð Þp , where
the indexes shown in parenthesis do not indicate a summation in j. As the unit
vector ej is collinear with gi, thus
gj ¼
ffiffiffiffiffiffiffiffi
g jjð Þ
p
ej ð1:9:4Þ
and with expression (1.9.4) in expression (1.9.3)
dx*j ¼ ffiffiffiffiffiffiffiffig jjð Þp dxj ð1:9:5Þ
In an analogous way, by means of expression (1.9.2)
u*k ¼
ffiffiffiffiffiffiffiffiffi
g kkð Þ
p
uk ð1:9:6Þ
The physical components u�k have the characteristics of displacement, so they
can be added vectorially (parallelogram rule), denoting the contravariant physical
components of the vector. These components are not unique. Let another variety of
components ũk that represents the projection of vector u on the direction of the unit
vector ek. Consider ẽk the reciprocal unit vector of ek, whereby, for this reciprocal
basis,
euk ¼ u � ek ð1:9:7Þ
u ¼ ukgk ¼ eukeek ð1:9:8Þ
but ẽk is collinear with g
k thus
ek ¼ 1eek
gk ¼
ffiffiffiffiffiffiffiffiffi
g kkð Þ
p
ek ¼
ffiffiffiffiffiffiffiffiffi
g kkð Þ
p
eek ) eek ¼
ffiffiffiffiffiffiffiffiffi
g kkð Þ
p
gk
eek ¼ ffiffiffiffiffiffiffiffiffig kkð Þp gk ð1:9:9Þ
where the indexes shown in parenthesis do not indicate summation in k, and with
expression (1.9.9) in expression (1.9.7)
uk ¼ ffiffiffiffiffiffiffiffiffig kkð Þp euk ð1:9:10Þ
The physical components ũk are the covariant components of vector u. Putting
uk ¼ g kkð Þuk and with the expressions (1.9.6) and (1.9.10) then in an orthogonal
coordinate system u*k ¼ euk. This shows that the distinction between the covariant
and contravariant basis disappears when the coordinate system is orthogonal.
1.8 Physical Components of a Tensor 63
Figure 1.11 shows the physical components of vector u in the curvilinear
coordinate system Xi. The components ukffiffiffiffiffiffiffi
g kkð Þ
p (expression (1.9.10)) represent the
lengths of the projections which are orthogonal to the coordinate axes of the
referential system. The components
ffiffiffiffiffiffiffiffiffi
g kkð Þ
p
uk (expression (1.9.6)) represent the
lengths the of the sides of the parallelepiped, which diagonal is the vector u.
Exercise 1.23 Calculate the contravariant, covariant, and physical components of
the velocity vector of a point vi ¼ dxidt , in terms of the cylindrical coordinates of the
point xi(r, θ, z).
Cartesian to cylindrical Cylindrical to Cartesian
x1 ¼ x1 cos x2 ¼ r cos θ
x1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x1
� � 2 þ x2� � 2q ¼ r
x2 ¼ x2 sin x2 ¼ r sin θ x2 ¼ arctg x2
x1
¼ θ
x3 ¼ x3 ¼ z x3 ¼ x3 ¼ z
The Cartesian coordinates of the vector are vi ¼ dxidt , and for the cylindrical
coordinates
vi ¼ ∂x
i
∂xj
vj ) vi ¼ ∂x
i
∂xj
dxi
dt
¼ dx
i
dt
This shows that the contravariant components of vector v are derivatives with
respect to the time of the position vector defined by the coordinates xi, then
vi
� � ¼ dx1
dt
,
dx2
dt
,
dx3
dt

 �
¼ dr
dt
,
dθ
dt
,
dz
dt

 �
For the covariant components in terms of the cylindrical coordinates
1X
2X
3X
P
1g
2g
3g
u
11
1
g
u
22
2
g
u
33
3
g
u
Fig. 1.11 Physical
components of the vector u
in the curvilinear coordinate
system Xi
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vi ¼ ∂x
i
∂xi
vj ¼ gijvj
gij ¼
1 0 0
0 r2 0
0 0 1
24 35
Developing the expression of vi
v1 ¼ g11v1 þ g12v2 þ g13v3 ¼ v1 ¼
dx1
dt
¼ dr
dt
v2 ¼ g12v1 þ g22v2 þ g23v3 ¼ x1
� �2 dx2
dt
¼ r2 dθ
dt
v3 ¼ g31v1 þ g23v2 þ g33v3 ¼
dx3
dt
¼ dz
dt
whereby
vi
� � ¼ dx1
dt
, x1
� �2 dx2
dt
,
dx3
dt

 �
¼ dr
dt
, r2
dθ
dt
,
dz
dt

 �
The vector norm is given by
vk k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dx1
dt
� �2
þ x1ð Þ2 dx
2
dt
 �2
þ dx
3
dt
� �2s
whereby for its physical components
vf g ¼ dx
1
dt
, x1
dx2
dt
,
dx3
dt

 �
¼ dr
dt
, r
dθ
dt
,
dz
dt

 �
Exercise 1.24 Let the vector u ¼ 3g1 þ g2 þ 2g3, having g1 ¼ 2e1, g2 ¼ 2e1 þ e2,
and g3 ¼ 2e1 þ e2 þ 3e3, where e1, e2, e3 are orthonormal vectors, calculate their
contravariant physical components.
From the covariant basis
g1 � g1 ¼ 2e1:2e1 ¼ 4 )
ffiffiffiffiffiffi
g11
p ¼ 2
g2 � g2 ¼ 2e1:2e1 þ e2 � e2 ¼ 4þ 1 ¼ 5 )
ffiffiffiffiffiffi
g22
p ¼
ffiffiffi
5
p
g3 � g3 ¼ 2e1:2e1 þ e2 � e2 þ 3e3 � 3e3 ¼ 4þ 1þ 9 ¼ 14 )
ffiffiffiffiffiffi
g33
p ¼
ffiffiffiffiffi
14
p
follows
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u*1 ¼ u1 ffiffiffiffiffiffig11p ¼ 3� 2 ¼ 6
u*2 ¼ u2 ffiffiffiffiffiffig22p ¼ 2� ffiffiffi5p ¼ 2 ffiffiffi5p
u*3 ¼ u3 ffiffiffiffiffiffig33p ¼ 1� ffiffiffiffiffi14p ¼ ffiffiffiffiffi14p
1.8.1.1 Physical Components of the Second-Order Tensor
The contravariant physical components of the vectors u and v are given by
expression (1.9.10)
ui ¼ ffiffiffiffiffiffiffiffig iið Þp eui ) eui ¼ uiffiffiffiffiffiffiffiffig iið Þp
vj ¼ ffiffiffiffiffiffiffiffig jjð Þp evj ) evj ¼ vjffiffiffiffiffiffiffiffig jjð Þp
For the second-order tensor
eTij ¼ euievj
eTij ¼ uiffiffiffiffiffiffiffiffi
g iið Þ
p vjffiffiffiffiffiffiffiffi
g jjð Þ
p ¼ 1ffiffiffiffiffiffiffiffi
g iið Þ
p ffiffiffiffiffiffiffiffi
g jjð Þ
p uivj
� � ¼ Tijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g iið Þg jjð Þ
p
Tij ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig iið Þg jjð Þp eTij ð1:9:11Þ
In a related manner, for the contravariant physical components
*Tij ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig iið Þg jjð Þp eTij ð1:9:12Þ
The obtaining of the physical components of tensors of a higher order follows
the analogous way to that of the second-order tensors.
1.9 Tests of the Tensorial Characteristics of a Variety
The transformation law of the tensors and the quotient law allow establishing a
group of functions Np of the coordinates of the referential system Xi which are the
components of a tensor. The tensorial nature of the functions that fulfill these
requirements is highlighted by the invariance that this variety has when there is a
change of the coordinate system. However, the evaluation if a variety has tensorial
characteristics by means of the quotient law is not wholly complete, as it will be
shown next applying to the group ofN2 components of a variety Tpq, for which it is
desired to search if it has the characteristics of a tensor. Multiplying Tpq by an
66 1 Review of Fundamental Topics About Tensors
arbitrary vector vp and admitting by hypothesis that Tpqv
pvq ¼ m, where m is a
scalar, it provides for a new coordinate system Tijv
ivj ¼ m, and as m is an invariant,
then m ¼ m, by means of the transformation law of vectors
Tpqv
pvq ¼ ∂x
i
∂xp
∂xj
∂xq
Tijv
pvq
Then
Tpq � ∂x
i
∂xp
∂xj
∂xq
Tij
� �
vpvq ¼ 0 ð1:10:1Þ
The summation rule is applied varying the indexes p and q, so the product vpvq is
not, in general, null. Consider the vectors vi with unit components 1, 0, 0� � �0ð Þ,
0, 1, 0� � �0ð Þ, and 0, 0, 0� � �1ð Þ, the term in parenthesis of expression (1.10.1) stays
T11 � ∂x
i
∂x1
∂xj
∂x1
Tij
� �
v1v1 ¼ 0
and as v1v1 6¼ 0
T11 � ∂x
i
∂x1
∂xj
∂x1
Tij ¼ 0 ð1:10:2Þ
In an analogous way it results in
T22 � ∂x
i
∂x2
∂xj
∂x2
Tij ¼ 0 ð1:10:3Þ
and so successively for the other values assumed for the indexes. This shows that
for p ¼ q the terms in parenthesis from expression (1.10.1) cancel each other.
However, for p 6¼ q the complementary analysis of this expression behavior
becomes necessary.
Let vector viwith components v1, v2, 0, � � �0ð Þ, whereby from expression (1.10.1)
for p, q ¼ 1, 2, it follows that
T11 � ∂x
i
∂x1
∂xj
∂x1
Tij
� �
v1v1 þ T12 � ∂x
i
∂x1
∂xj
∂x2
Tij
� �
v1v2
þ T21 � ∂x
i
∂x2
∂xj
∂x1
Tij
� �
v2v1 þ T22 � ∂x
i
∂x2
∂xj
∂x2
Tij
� �
v2v2 ¼ 0
ð1:10:4Þ
Expressions (1.10.2) and (1.10.3) simplify expression (1.10.4), for the coeffi-
cients of the terms vpvq are null for p ¼ q. For p 6¼ q with Tij ¼ Tji
1.9 Tests of the Tensorial Characteristics of a Variety 67
∂xi
∂x1
∂xj
∂x2
Tij ¼ ∂x
i
∂x2
∂xj
∂x1
Tij
and with the hypothesis of symmetry results in
∂xi
∂x1
∂xj
∂x2
Tij ¼ ∂x
i
∂x2
∂xj
∂x1
Tji
Expression (1.10.4) is rewritten as
T12 þ T21ð Þ � Tij þ Tji
� � ∂xi
∂x1
∂xj
∂x2
 �
v1v2 ¼ 0
and as the components v1 and v2 are arbitrary, for v1 ¼ v2 ¼ 1
T12 þ T21 ¼ Tij þ Tji
� � ∂xi
∂x1
∂xj
∂x2
ð1:10:5Þ
Generalizing expression (1.10.5) for the variation of the indexes
p, q ¼ 1, 2, 3, . . ., it results in
Tpq þ Tqp ¼ Tij þ Tji
� � ∂xi
∂xp
∂xj
∂xq
ð1:10:6Þ
Expression (1.10.6) is the transformation law of second-order tensors, for the
term Tpq þ Tqp
� �
represents the symmetric part of tensor 2Tpq. However, the
antisymmetric part of this tensor is not contained in this analysis, whereby it cannot
be concluded that this portion has tensorial characteristics. It is concluded that only
the symmetric part of the N2 components of variety Tpq is a tensor, for when
applying the quotient law to this portion it transforms according to the transforma-
tion law of second-order tensors. This is the reason why the quotient law must be
applied with caution, so as to avoid evaluation errors when checking the tensorial
characteristics of a variety.
The transformation law of tensors and the consideration of invariance of the
variety when having a linear transformation form the criterion that is most appro-
priate to evaluate if the Np components of this variety have tensorial characteristics.
Problems
1.1 Use the index notation to write:
(a)
dx1
dt
dx2
dt
8><>:
9>=>; ¼ a11 a12a21 a22
 �
x1
x2

 �
; (b) Φ ¼ x21 þ x22 þ 2x1x2
Answer: (a) x, t ¼ aijxj; (b) Φ ¼ xixj.
68 1 Review of Fundamental Topics About Tensors
1.2 Let aij constant 8i, j, calculate ∂ aijxixjð Þ∂xk ) where aij ¼ aji.
Answer: ∂ 2aikxið Þ∂x‘ ¼ 2aik
1.3 If aijkx
ixjxk ¼ 0 8x1, x2, � � �, xn and aijk are constant values, show that
akji þ ajki þ aikj þ aijk þ akij þ ajik ¼ 0.
1.4 Calculate for i, j ¼ 1, 2, 3: (a) δijAi, (b) δijAij, (c) δii, (d) δijδji, (e) δijδjkδk‘,
(f) C ¼ aijkaijk
Answer: (a) δijAi ¼ Aj, (b) δijAi ¼ Aii ¼ Ajj, (c) δi i ¼ 3, (d) δijδijijji jiji ¼ 3,
(e) δijδ
ijij
jkδ
jk jk
k‘¼k‘k‘δi‘, and (f) 64.
1.5 Calculate the Jacobian of the linear transformations between the coordinate
systems (a) x1 ¼ x1; x2 ¼ x1x2; x3 ¼ x1x2x3 ; (b) x1 ¼ x1 cos x2 sin x3; x2 ¼
x1 sin x2 sin x3; x3 ¼ x1 cos x3.
Answer: (a) J ¼ x1� �2x2; (b) J ¼ � x1� � 2 sin x3.
1.6 Given the tensor Tk‘ ¼
1 0 0
0 2 1
0 1 3
24 35 in the coordinate system Xi, calculate the
components of this tensor in the coordinate system X
i
, with the relations
between the coordinates of these systems given by
x1 ¼ x1 þ x3, x2 ¼ x1 þ x2, x ¼ x3.
Answer: Tij ¼
3 2 2
2 2 1
2 1 4
24 35
1.7 Given the tensor Tij ¼
1 1 5
1 2 �1
5 �1 3
24 35 in the coordinate system Xi, calculate
the components of this tensor in the coordinate system X
i
, with the relations
between the coordinates of these systems given by
x1 ¼ x1 þ 2x2, x2 ¼ 3x3, x ¼ x3.
Answer: T
ij ¼
25 8 2
8 4 10
2 10 3
24 35
1.8 Show that (a) tr Tð Þ ¼ tr Sð Þ, where T and S are, respectively, a symmetric and
an antisymmetric tensor, both of the second order; (b)Tijk‘ ¼ 0, being Tijk‘ one
symmetric tensor in the indexes i, j and antisymmetric in the indexes j, ‘.
1.9 Decompose the second-order tensor in two tensors, one symmetric and
another antisymmetric
1.9 Tests of the Tensorial Characteristics of a Variety 69
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Tij ¼
�1 2 0
3 0 �2
1 0 1
24 35
Answer:
�1 2:5 0:5
2:5 0 �1
0:5 �1 1
24 35 0 �0:5 �0:50:5 0 �1
0:5 1 0
24 35.
1.10 Consider the tensor Tij that satisfies the tensorial equation mTij þ nTji ¼ 0,
wherem > 0 and n > 0 are scalars. Prove that if Tij is a symmetric tensor, then
m ¼ �n, and m ¼ n if this is an antisymmetric tensor.
1.11 Let the Cartesian coordinate system with basis vectors e1, e2, e3. Calculate the
metric tensor of the space with basis vectors g1 ¼ e1, g2 ¼ e1 þ e2, and
g3 ¼ e1 þ e2 þ e3.
Answer: gij ¼
1 1 1
1 2 2
1 2 3
24 35
1.12 Let the basis vectors e1, e2 of the coordinate system X
i with metric tensor gij
and the basis vectors ee1 ¼ 3e1 þ e2 and ee2 ¼ �e1 þ 2e2 of the coordinate
system eXi. Calculate the covariant components of the metric tensoregij in terms
of the components of gij.
Answer: eg11 ¼ 9g11 þ 6g12 þ g22 ; eg12 ¼ eg21 ¼ �3g11 þ 5g12 þ 2g22 ;eg22 ¼ g11 � 4g12 þ 4g22.
1.13 Calculate the contravariant components of the vector u ¼ g1 þ 2g2 þ g3,
where the covariant base vectors are g1 ¼ e1, g2 ¼ e1 þ e2, g3 ¼ e3, being
e1, e2, e3 base vectors.
Answer: u ¼ 4g1 þ 7g2 þ 8g3.
1.14 Let the contravariant base vectors g1 ¼ e1; g2 ¼ e1 þ e2; and
g3 ¼ e1 þ e2 þ e3, where e1, e2, e3 are the vectors of the one orthonormal
base. Calculate the:
(a) Vectors g1, g2, g3 of the contravariant base
(b) Metric tensor and the conjugated tensor.
Answer: (a)
g1 ¼ e1 � e2
g2 ¼ e2 � e3
g3 ¼ e3
8<: ; (b) gij ¼
1 1 1
1 2 2
1 2 3
24 35 gij ¼ 2 �1 0�1 2 2
0 �1 1
24 35
1.15 Consider the coordinate system x1 ¼ x1 cos x2, x2 ¼ x1 sin x2, x3 ¼ x3. Calcu-
late the arc length along the parametric curve x1 ¼ a cos t, x2 ¼ a sin t, x3 ¼ b
t in the interval 0 	 t 	 c, being a, b, c positive constants.
Answer: L ¼ c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ b2
p
1.16 Calculate the angle between the vectors
(a) u 2;�3; 1ð Þ, v 3;�1;�2ð Þ; (b) u 2; 1;�5ð Þ, v 5; 0; 2ð Þ.
Answer: (a) 60o; (b) 90o
1.17 With i, j, k ¼ 1, 2, 3 calculate the following expressions:
(a) uivjδji � vkuiδki; (b) δijδji; (c) eijkuiujuk
70 1 Review of Fundamental Topics About Tensors
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Answer: (a) zero, (b) 3, (c) zero
1.18 Show that the followings expressions are invariants (a) Tijuivj; (b) Tii;
(c) det Tij.
1.19 Let the vector (a) ui show that if A
ijuiuj ¼ Bijuiuj, then Aij þ Aji ¼ Bij þ Bij;
(b) ui and if A
ijuiuj is invariant, show that A
ij þ Aji� � is a tensor.
1.20 Let Tpqrs an absolute tensor, show that if Tijk‘ þ Tij‘k ¼ 0 in the coordinate
system Xi, then Tijk‘ þ Tij‘k ¼ 0 in another coordinate system Xi.
1.21 Let the vector u ¼ g1 þ 2g2 þ g3, having g1 ¼ e1, g2 ¼ e1 þ e2, and
g3 ¼ e1 þ e2 þ e3, where e1, e2, e3 are orthonormal vectors, calculate their
contravariant physical components.
Answer: u*1 ¼ 1, u*2 ¼ 2 ffiffiffi2p , u*3 ¼ ffiffiffi3p1.22 Calculate the value of the permutation symbol e321546.
Answer: e321546 ¼ 1.
1.23 Show that
(a) eijkejki ¼ 6; (b) eijkujuk ¼ 0; (c) er‘merst ¼ δrstr‘m ¼
1
0
�1
8<: r, ‘, m, s, t ¼ 1,
2, 3;
(d) er‘me
rst ¼ δ s‘δ tm � δ t‘δ sm; (e) eijeij ¼ 2!; (f) eijk‘eijk‘ ¼ 4!; (g) eijk‘���eijk‘��� ¼ N!
where N is the index number.
1.24 For i, j, k, ‘ ¼ 1, 2, 3 show that
(a) δijk‘ ¼
δ ik δ
i
‘
δ jk δ
j
‘
				 				 ¼ δ ikδ j‘ � δ i‘δ jk, (b) δijkrst ¼ δ
i
r δ
i
s δ
i
t
δ jr δ
j
s δ
j
t
δ kr δ
k
s δ
k
t
						
						 ¼ δijk‘, (c) δijkijk ¼ 6.
1.25 Calculate the determinant by means of the expansion of the permutation
symbol
1 1 0 1
�1 0 1 1
0 1 �1 1
1 �1 1 0
								
								 ¼ eijk‘a1ia2ja3ka4‘
Answer: �3.
1.26 Show that (a) εijk ¼ ej � ek
� � � ei where ei, ej, ek are unit vectors of one
coordinate system; (b) if ui and u
i are associate tensors and ui ¼ εijkujvk,
then ui ¼ εijkujvk; (c) εijkuivjwk ¼ εijkuivjwk.
1.27 Simplify the expression F ¼ εijkεpqrAipAjqAkr.
1.28 Verify if the expression εmnpεmij þ εmnjεmpi ¼ εmniεmpj is correct or false.
Justify the answer.
1.29 Write the tensor components Tij ¼ 1
2
εijkAk‘ with i, j, k, ‘ ¼ 1, 2, 3, 4, and
show that if ijk‘ is an even permutation for the pair of 1234, then Tij ¼ Ak‘.
1.9 Tests of the Tensorial Characteristics of a Variety 71
Chapter 2
Covariant, Absolute, and Contravariant
Derivatives
2.1 Initial Notes
The curve represented by a function ϕ(xi) in a closed interval is continuous if this
function is continuous in this interval. If the curve is parameterized, i.e., ϕ[xi(t)]
being t2 a; b½ �, then it will be continuous if xi(t) are continuous functions in this
interval, and it will be smooth if it has continuous and non-null derivatives for a
value of t2 a; b½ �. The smooth curves do not intersect, i.e., the conditions
xi að Þ ¼ xi bð Þ will only be satisfied if a ¼ b. This condition defines a curve that
can be divided into differential elements, forming curve arcs. For the case in which
the initial and final points coincide, expressed by condition a ¼ b, the curve is
closed. The various differential elements obtained on the curve allow calculating its
line integral.
The curves can be smooth by part, i.e., they are composed of a finite number of
smooth parts (arc elements), connected in their initial and final point. This kind of
curve can intersect in one or more points, and if their extreme points coincide, it is
called a closed curve.
The differentiation condition of a function is associated to the concept of
neighborhood and limit. The neighborhood of a point P(xi) is defined admitting
that the very small radius ε, with which a sphere is traced, is centered on it. The
interior of this sphere is this point’s neighborhood of radius ε. This definition is
valid in the plane, changing the sphere for a circle, and is complemented admitting a
set of points, which is called an open set. The points interior to the cube shown in
Fig. 2.1 form an open set, for in each point P(xi) a sphere of radius ε can be drawn in
its interior, which will be contained in the cube’s interior. If the cube’s edges are
included, the result is a closed set.
© Springer International Publishing Switzerland 2016
E. de Souza Sánchez Filho, Tensor Calculus for Engineers and Physicists ,
DOI 10.1007/978-3-319-31520-1_2
73
2.2 Cartesian Tensor Derivative
The tensors derivative study begins with the research of what happens when a
scalar function is differentiated. The behavior of this kind of function leads to
the study of more general cases, such as those of the vectorial and tensorial
functions.
Consider a scalar function ϕ(xi) defined in a coordinate system Xi, which
derivative with respect to the variable xi is given by
∂ϕ xið Þ
∂xi
¼ ϕ, i ¼ Gi
and in another coordinate system X
i
has as derivative
Gi ¼ ∂ϕ x
ið Þ
∂xi
¼ ∂ϕ x
ið Þ
∂xi
¼ ∂ϕ x
ið Þ
∂xi
∂xi
∂xi
) Gi ¼ ∂x
i
∂xi
Gi
whereby
ϕ, i ¼
∂ϕ xið Þ
∂xi
ϕ, i
that is the transformation law of vectors, so
∂ϕ xið Þ
∂xi is a vector and defines the gradient
of the scalar function. In vectorial notation, it is graphed as u ¼ gradϕ xið Þ.
ε
P
Fig. 2.1 Neighborhood
of a point
74 2 Covariant, Absolute, and Contravariant Derivatives
Differentiating the scalar function again with respect to the variable xj results in
ϕ, ij ¼
∂xk
∂xj
∂xm
∂xi
� �
∂2ϕ
∂xm∂xk
and for i ¼ j
ϕ, ii ¼
∂xk
∂xi
∂xm
∂xi
� �
∂2ϕ
∂xm∂xk
δkm ¼ ∂x
k
∂xi
∂xm
∂xi
ϕ, ii ¼ δkm
∂2ϕ
∂xm∂xk
¼ ∂
2ϕ
∂xk∂xk
then ϕ(xi),ii is a scalar.
2.2.1 Vectors
In the study of vectors, there are two distinct manners to carry out a derivative: the
derivative of a vector and the derivative of a point. In this text, a few conceptual
considerations are made before calculating the derivative of a vector.
Consider the vectorial space EN in which the scalar variable t2 a; bð Þ is defined,
where for each value of this variable limited in the open interval there is a vector u
(t) embedded in this space which has a metric tensor. Let the vector u(t) defined by a
continuous function of the variable t, and that admits continuous derivatives, then
by means of an elementary increment Δt this vector will have an elementary
increment
Δu tð Þ ¼ u tþ Δtð Þ � u tð Þ
and when Δt ! 0 a vector v will exist such that
lim
Δu tð Þ
Δt
� v
� �
! 0
then the vector v is derived from the vector u(t) by means of the variation of the
parameter t, whereby
v ¼ du
dt
2.2 Cartesian Tensor Derivative 75
The rules of Differential Calculus are applicable to this kind of differentiation.
The other method of calculating a vector by means of a derivative is associated to
the concept of punctual space EN, with respect to which a variable t2 a; bð Þ is
associated in a univocal way to an arbitrary point P(t). Taking a fixed point
O contained in EN as reference origin, the vector r tð Þ ¼ OP tð Þ is determined.
Let the vector r(t) defined by a continuous and derivable function, then the result
is the vector
dr tð Þ
dt that does not depend on the origin O, but only on the point P(t).
This statement can be demonstrated admitting a new arbitrary point O* for which
the result is the vector
OP ¼ OO* þO*P
Fixing the vector OO* and calculating the derivative of these vectors with
respect to the variable t, the result is
d OPð Þ
dt
¼ d OO
*
� �
dt
þ d O
*P
� �
dt
) d OPð Þ
dt
¼ d O
*P
� �
dt
This equality proves the independence of the vector
dr tð Þ
dt with respect to the
arbitrated origin. This vector is derived from the point P(t), and in the case of the
points defining a smooth curve C contained in the space EN, continuous and
differentiable, and will be tangent to the curve in each point for which this
derivative was calculated.
In the general case of the vector being a function of various scalar variables
r xið Þ, i ¼ 1, 2, . . . ,N, the result is by means of the differentiation rules of Differ-
ential Calculus
∂2r tð Þ
∂xi∂xk
¼ ∂
2
r tð Þ
∂xk∂xi
For a vectorial function of scalar variables, the vector derivative is given by
dr tð Þ ¼ ∂r tð Þ
∂xi
dxi
If the curve C is a function of the variables xi, and these depend on the variable
t2 a; bð Þ, i.e., xi ¼ xi tð Þ, the curve is parameterized, then the hypothesis of Differ-
ential Calculus is applicable, and the total differential of the vector r[xi(t)] is
dr tð Þ
dt
¼ ∂r tð Þ
∂xi
dxi
dt
The rules applicable to the differentiation of vectors, and to the vectors obtained
by means of differentiation of a point are the same. The concept of vector calculated
by differentiation of one point can be extended to the study of tensors, where the
points to be analyzed are contained in the tensorial space EN.
76 2 Covariant, Absolute, and Contravariant Derivatives
Exercise 2.1 Show that derivative of the vector r with constant direction maintains
its direction invariable.
Let
r tð Þ ¼ ψ tð Þu
where ψ(t) is a parameterized scalar function and u is a constant vector, thus
dr tð Þ
dt
¼ ψ 0 tð Þuþ ψ tð Þ du
dt
¼ ψ 0 tð Þu
then r and
dr tð Þ
dt have thesame direction.
2.2.2 Cartesian Tensor of the Second Order
The tensorial functions defined by Cartesian tensors are often found in applications
of physics and the areas of engineering. Let, for instance, the derivative of this kind
of tensorial function that is a particular case of the derivative of tensors expressed in
curvilinear coordinate systems. For the analysis of this derivative a Cartesian tensor
of the second order Tij is admitted, which components are functions of the coordi-
nates xi. The tensor with this characteristic is a function of the point considered in
the space E3. The transformation law of the tensors of the second order is given by
Tij ¼ ∂x
p
∂xi
∂xq
∂xj
Tpq
For the Cartesian coordinate systems the coefficients ∂x
p
∂xi
and ∂x
q
∂xj
are constants, for
they represent the variation rates for the linear transformations, so they do not
depend on the point’s coordinates. The derivative of this expression is given by
∂Tij
∂x‘
¼ ∂x
p
∂xi
∂xq
∂xj
∂Tpq
∂xm
∂xm
∂x‘
� �
) ∂Tij
∂x‘
¼ ∂x
p
∂xi
∂xq
∂xj
∂xm
∂x‘
∂Tpq
∂xm
that is the transformation law of tensors of the third order, concluding that the
derivative of tensor Tij increased the order of this tensor in one unit. This conclusion
is general and applicable to any Cartesian tensor.
This kind of derivative is not valid for the more general tensors. The concept of
tensors derivative must, therefore, be generalized for tensors which components are
given in curvilinear coordinate systems.
Exercise 2.2 Show that if Tij is a Cartesian tensor of the second order, then
∂2Tij
∂xk∂xm
will be a tensor of the fourth order.
2.2 Cartesian Tensor Derivative 77
The tensor is a function of the coordinates Tij(x1, x2, x3), whereby the result is
∂xi
∂xj
¼ δij ) ∂
2
xi
∂xj∂xk
¼ 0
and by transformation law
Tpq ¼ ∂xi∂xp
∂xj
∂xq
Tij
then
∂2Tpq
∂xr∂xs
¼ ∂
2
∂xr∂xs
∂xi
∂xp
∂xj
∂xq
Tij
� �
¼ ∂xk
∂xr
∂
∂xk
∂xm
∂xs
∂
∂xm
∂xi
∂xp
∂xj
∂xq
Tij
� �
 �
¼ ∂xk
∂xr
∂xm
∂xs
∂xi
∂xp
∂xj
∂xq
∂
∂xk
∂Tij
∂xm
� �
In a mnemonic manner
Tpq, rs ¼ ∂xk∂xr
∂xm
∂xs
∂xi
∂xp
∂xj
∂xq
Tij,mk
that is the transformation law of tensors of the fourth order as Tij,mk ¼ Tij,km the
result is that Tij,km ¼ ∂
2
Tij
∂xk∂xm
is a tensor of the fourth order.
2.3 Derivatives of the Basis Vectors
Consider the contravariant vector uk ¼ uk xið Þ defined in terms of the parametric
curve xi ¼ xi tð Þ, expressed with respect to the Cartesian coordinate system Xi. By
means of the transformation law of vectors, the result for the curvilinear coordinate
system X
i
is
u‘ ¼ ∂x
i
∂xk
uk
and with the techniques of differentiation with respect to the parameter t results in
du‘
dt
¼ ∂x
i
∂xk
duk
dt
þ ∂
2
xi
∂xk∂x‘
dx‘
dt
uk ð2:3:1Þ
78 2 Covariant, Absolute, and Contravariant Derivatives
that only represents a contravariant vector if, and only if, xi is a linear function of xk.
The first term on the right of this expression represents an ordinary differentiation
of a vectorial function expressed in Cartesian coordinates, but the second term
contains the derivatives curvilinear coordinates xi, relative to a coordinate system
that varies as a function of the points of the space.
The study of this term is carried out considering the Cartesian coordinate system
Xi and the curvilinear coordinate system X
i
, with unit vectors ei and gi, respectively
(Fig. 2.2). Whereby defining the position vector r of point P with respect to the
coordinate system Xi by means of their contravariant components
r ¼ xiei
the differential total of this vector is given by
dr ¼ ∂r
∂xi
dxi ð2:3:2Þ
As the basis vectors ei do not depend on point P:
dr ¼ dxiei
so
ei ¼ ∂r∂xi ð2:3:3Þ
O
P
1
X
2
X
3
X
1
X
2
X
3
X
3
g
1
g
2
g
2
e
3
e
1
e
Fig. 2.2 Cartesian Xi and
curvilinear X
i
coordinates
with basis vectors ei and gi,
respectively
2.3 Derivatives of the Basis Vectors 79
With respect to the local coordinate systemX
i
the result is the differential total of
the position vector r:
dr ¼ r, i dxi ð2:3:4Þ
Whereby the base vectors of the curvilinear coordinate system results
gi ¼ r, i ð2:3:5Þ
that shows that the unit vectors gi are tangent to the curves that define the
curvilinear coordinate system X
i
, that varies for each point of the vectorial space
E3, and as the unit vectors ei do not vary
∂r
∂xk
¼ ∂x
i
∂xk
ei
Comparing with expression (2.3.5)
gk ¼
∂xi
∂xk
ei ð2:3:6Þ
then
ei ¼ ∂x
j
∂xi
gj ð2:3:7Þ
The covariant derivative of the base vector defined by the expression (2.3.6) is
given by
gk, ‘ ¼
∂2xi
∂xk∂x‘
ei ð2:3:8Þ
and substituting expression (2.3.7) in this expression
gk, ‘ ¼
∂xj
∂xi
∂2xi
∂xk∂x‘
gj
Defining the variety
Γ jk‘ ¼
∂xj
∂xi
∂2xi
∂xk∂x‘
ð2:3:9Þ
with which the covariant derivatives of the basis vectors of the curvilinear coordi-
nate system can be written as linear combination of the base vector gj:
gk, ‘ ¼ Γ jk‘gj ð2:3:10Þ
80 2 Covariant, Absolute, and Contravariant Derivatives
2.3.1 Christoffel Symbols
The coefficients determined by expression (2.3.9) can be expressed in terms of the
derivatives of the metric tensor and its conjugated tensor. For the derivatives of
the contravariant basis vectors considering another variety Γ
i
jm:
gi, j ¼ Γ ijmgm ð2:4:1Þ
Writing
gi � gj
� �
, k
¼ δ ij
� �
, k
¼ g i, k � gj þ gj , k � gi ¼ 0 ð2:4:2Þ
and substituting the expressions (2.3.10) and (2.4.1) in expression (2.4.2)
Γ
i
jkg
k � gj þ Γ ijkgj � gi ¼ 0 ð2:4:3Þ
then
Γ
i
jk ¼ �Γ ijk
and with the expressions (2.3.9), (2.3.10), (2.4.1) and with the prior expression it
follows that
Γmij ¼
∂xm
∂xk
∂2xk
∂2xi∂xj
gi, j ¼ Γmij gm
gi, j ¼ �Γ ijmgm ð2:4:4Þ
The relation between the covariant and contravariant unit vectors is defined by
gi ¼ gijgj
and the derivative of this expression with respect to the coordinate xk is given by
gi, k ¼ gij,kgj þ gijgj, k
Replacing expressions (2.3.10) and (2.4.4) in this last expression
Γmikgm ¼ gij,kgj � gijΓ jkmgm
and with the multiplying by gn
2.3 Derivatives of the Basis Vectors 81
Γmikgm � gn ¼ gij,kgj � gn � gijΓ jkmgm � gn
Γmik δ
n
m ¼ gij,kgjn � gijΓ jkmgmn
Γ nik ¼ gij,kgjn � gijgmnΓ jkm
The multiplying of this last expression by gnp provides
gnpΓ
n
ik þ gijgmngnpΓ jkm ¼ gij,kgjngnp ) gnpΓ nik þ gijδmp Γ jkm ¼ gij,kδ jp
whereby
gnpΓ
n
ik þ gijΓ jkp ¼ gip,k ð2:4:5Þ
and with the cyclic permutation of the free indexes i, p, k of expression (2.4.5)
gnkΓ
n
pi þ gpjΓ jik ¼ gpk, i ð2:4:6Þ
gniΓ
n
kp þ gkjΓ jpi ¼ gki,p ð2:4:7Þ
Multiplying expression (2.4.5) by �1/2 and expressions (2.4.6) and (2.4.7) by
1/2 and adding
� 1
2
gnpΓ
n
ik þ gijΓ jkp
� �
þ 1
2
gnkΓ
n
pi þ gpjΓ jik
� �
þ 1
2
gniΓ
n
kp þ gkjΓ jpi
� �
¼ 1
2
gpk, i þ gki,p � gip,k
� �
and with the change of the index n for the index j, and considering the symmetry of
the metric tensor
gkjΓ
j
ip ¼
1
2
gpk, i þ gki,p � gip,k
� �
The term to the right of the expression shows the existence of coefficients that
are functions only of the partial derivatives of the metric tensor that define the
Christoffel symbol of first kind
p; k½ � ¼ Γip,k ¼ 1
2
gpk, i þ gki,p � gip,k
� �
¼ 1
2
∂gpk
∂xi
þ ∂gki
∂xp
� ∂gip
∂xk
� �
ð2:4:8Þ
Multiplying expression (2.4.8) by gkm:
gkmgkjΓ
j
ip ¼
1
2
gkm gpk, i þ gki,p � gip,k
� �
) δmj Γ jip ¼
1
2
gkm gpk, i þ gki,p � gip,k
� �
82 2 Covariant, Absolute, and Contravariant Derivatives
whereby
m
ip

 �
¼ Γmip ¼
1
2
gkm gpk, i þ gki,p � gip,k
� �
¼ 1
2
gkm
∂gpk
∂xi
þ ∂gki
∂xp
� ∂gip
∂xk
� �
ð2:4:9Þ
The term to the right of this expression shows the existence of coefficients that
depend on the partial derivatives of the metric tensor and the conjugate metric
tensor. The coefficients represented by Γik‘ given by expressions (2.3.9) and (2.4.9)
define Christoffel symbol of second kind. Expression (2.4.9) is more convenient for
calculating these coefficients than expression (2.3.9).
Multiplying the terms of expression (2.3.10) by gi:
gi � gk, ‘ ¼ Γ jk‘gi � gj ¼ δ ijΓ jk‘
Γ ik‘ ¼ gi � gk, ‘ ð2:4:10Þ
2.3.2 RelationBetween the Christoffel Symbols
Expressions (2.4.8) and (2.4.9) relate the two Christoffel symbols, i.e.:
Γmij ¼ gkmΓij,k ð2:4:11Þ
then the Christoffel symbol of second kind is the raising of the third index of the
Christoffel symbol of first kind.
Expression (2.4.10) written in terms of the Christoffel symbol of first kind is
given by
Γ kij ¼ gkpΓij, p ¼ gk � gi, j
and multiplying the members by gkp
gkpg
kpΓij, p ¼ gkpgk � gi, j ) Γij, p ¼ gkpgk � gi, j ) Γij, p ¼ gk � gp � gk � gi, j
then
Γij, p ¼ gp � gi, j ð2:4:12Þ
2.3 Derivatives of the Basis Vectors 83
2.3.3 Symmetry
For the Christoffel symbol of first kind
Γij,k ¼ 1
2
∂gjk
∂xi
þ ∂gik
∂xj
� ∂gij
∂xk
� �
Γji,k ¼ 1
2
∂gik
∂xj
þ ∂gjk
∂xi
� ∂gji
∂xk
� �
and considering the symmetry of the metric tensor gik ¼ gki, gjk ¼ gkj, gij ¼ gji it
results in
Γij,k ¼ Γji,k
then the Christoffel symbol of first kind is symmetrical with respect to the first two
indexes, and with considering this symmetry results to the Christoffel symbol of
second kind
Γmij ¼ gkmΓij,k ¼ gkmΓji,k ¼ Γmji
that is symmetrical in regard to a permutation of the lower indexes.
2.3.4 Cartesian Coordinate System
For the Cartesian coordinate systems the elements of the metric tensor are gij ¼ δij,
whereby for p ¼ 1, 2, . . . ,N it results in
∂gip
∂xp
¼ 0
By means of the definition of the Christoffel symbol of first kind
Γij,k ¼ 1
2
∂gjk
∂xi
þ ∂gik
∂xj
� ∂gij
∂xk
� �
¼ 0
It is verified that the Christoffel symbol of second kind is cancelled, for
Γ pij ¼ gpkΓij,k ¼ 0
then for the Cartesian, orthogonal, or oblique coordinate systems, all the terms of
Γij,k and Γ
p
ij are null.
84 2 Covariant, Absolute, and Contravariant Derivatives
2.3.5 Notation
The oldest notations for the Christoffel symbols are
ij
k
 �
for the symbol of first
kind, and
ij
k

 �
for the symbol of second kind. Improving this notations Levi-Civita
adopted the spelling [ij, k] and {ij, k}, which second symbol was later improved by
various authors to
k
ij

 �
, where the indexes were placed in more adequate and
logical positions. This notation is well adopted, using the representation [ij, k] for
the Christoffel symbol of first kind.
Hermann Weyl used the Greek letter Γ to denote these symbols, which positions
of the indexes indicates the kind it represents: Γij,k and Γkij. This symbology is
known as the notation of the Princeton School. A few authors invert the position of
the indexes and write Γk,ij.
The argument for adopting the notations [ij, k] and
k
ij

 �
is that the Princeton
School notation leads to confusing these coefficients with a tensor. However, this
argument does not make its adoption valid, for the use of brackets or keys could also
lead to confusion with a matrix or column matrix. That is not the case. The use of
the notations Γij,k and Γkij, even not being universally accepted, has in its favor the
economy of characters in a text with many expressions containing these symbols.
Several authors do not use the comma for indicating the differentiation with respect
to one of the indexes in the Christoffel symbol of first kind, and write Γijk.
2.3.6 Number of Different Terms
For the tensorial space EN where i, j ¼ 1, 2, . . . ,N, it is verified that the metric
tensor gij has N
2 terms
gij ¼
g11 g12 � � � g1N
g21 g22 � � � g2N
⋮ ⋮ ⋮ ⋮
gN1 gN2 � � � gNN
2664
3775
N�N
This matrix has N diagonal terms gii, so N
2 � N� � terms remain in their sides. As
gij is symmetrical, the result is
1
2
N2 � N� � terms for i 6¼ j. The total of different
terms in the metric tensor is N2 þ 1
2
N2 � N� � ¼ N Nþ1ð Þ
2
.
For each kind of Christoffel symbol N derivatives of gij are calculated, so the
number of different terms for these coefficients is given by
N N2þ1ð Þ
2
.
2.3 Derivatives of the Basis Vectors 85
2.3.7 Transformation of the Christoffel Symbol of First Kind
Let Γpq,r defined in the coordinate system X
i and Γij,k expressed in the coordinate
systemX
i
, then using the expression that defines the Christoffel symbol of first kind,
and adopting the notation gij,k ¼ ∂gij∂xk it follows that
gij,k ¼
∂
∂xk
gpq
∂xp
∂xi
∂xq
∂xj
� �
¼ ∂gpq
∂xk
∂xp
∂xi
∂xq
∂xj
þ gpq
∂2xp
∂xk∂xi
∂xq
∂xj
þ gpq
∂xp
∂xi
∂2xq
∂xk∂xj
ð2:4:13Þ
By the chain rule
∂gpq
∂xk
¼ ∂gpq
∂xr
∂xr
∂xk
¼ gpq, r
∂xr
∂xk
ð2:4:14Þ
gij,k ¼ gpq, r
∂xp
∂xi
∂xq
∂xj
∂xr
∂xk
þ gpq
∂2xr
∂xk∂xi
∂xs
∂xj
 !
þ gpq
∂2xs
∂xk∂xj
∂xr
∂xi
 !
ð2:4:15Þ
and with cyclic permutation of the indexes in each term of the previous expression
and with gqp ¼ gpq
gjk, i ¼ gqr,p
∂xq
∂xj
∂xr
∂xk
∂xp
∂xi
þ gpq
∂2xq
∂xi∂xj
∂xp
∂xk
 !
þ gpq
∂2xp
∂xi∂xk
∂xq
∂xj
 !
ð2:4:16Þ
gki, j ¼ grp,q
∂xr
∂xk
∂xp
∂xi
∂xq
∂xj
þ gpq
∂2xp
∂xj∂xk
∂xq
∂xi
 !
þ gpq
∂2xp
∂xj∂xi
∂xq
∂xk
 !
ð2:4:17Þ
Γij,k ¼ 1
2
∂gjk
∂xi
þ ∂gki
∂xj
� ∂gij
∂xk
� �
ð2:4:18Þ
By substitution
Γij,k ¼ ∂x
p
∂xi
∂xq
∂xj
∂xr
∂xk
Γpq, r þ gpq
∂2xp
∂xi∂xj
∂xq
∂xk
ð2:4:19Þ
that is the transformation law of the Christoffel symbol of first kind. The second
term to the right of this expression shows that these coefficients are not the
components of a tensor.
86 2 Covariant, Absolute, and Contravariant Derivatives
2.3.8 Transformation of the Christoffel Symbol
of Second Kind
Writing the Christoffel symbol of second kind in terms of the components of a new
coordinate system X
i
:
Γ
i
jk ¼ gipΓjk,p ¼ gqr
∂xj
∂xq
∂xp
∂xr
� �
Γjk,p ð2:4:20Þ
As the Christoffel symbol of first kind transforms by mean of expression (2.4.19)
Γjk,p ¼ Γ‘m,n ∂x
‘
∂xj
∂xm
∂xk
∂xn
∂xp
þ g‘m
∂2x‘
∂xj∂xk
∂xm
∂xp
ð2:4:21Þ
and substituting expression (2.4.21) in expression (2.4.20) it follows that
Γ
i
jk ¼ gqr Γrn
∂x‘
∂xj
∂xm
∂xk
∂xn
∂xp
∂xi
∂xq
∂xp
∂xr
� �
þ gqrg‘m
∂2x‘
∂xj∂xk
∂xm
∂xp
∂xi
∂xq
∂xp
∂xr
 !
Γ
i
jk ¼ gqr Γ‘m,nδnr
∂x‘
∂xj
∂xm
∂xk
∂xi
∂xq
þ gqrg‘mδmr
∂2x‘
∂xj∂xk
∂xi
∂xq
 !
Γ
i
jk ¼ gqrΓ‘m,nδnr
∂x‘
∂xj
∂xm
∂xk
∂xi
∂xq
þ gqrg‘mδmr
∂2x‘
∂xj∂xk
∂xi
∂xq
Γ
i
jk ¼ gqrΓ‘m,nδnr
∂x‘
∂xj
∂xm
∂xk
∂xi
∂xq
þ gqrg‘r
∂2x‘
∂xj∂xk
∂xi
∂xq
gqrΓ‘m, r ¼ Γ q‘m
gqrg‘r ¼ δq‘
Γ
i
jk ¼ Γ q‘m
∂x‘
∂xj
∂xm
∂xk
∂xi
∂xq
þ ∂
2
x‘
∂xj∂xk
∂xi
∂xq
Replacing the dummy indexes ‘ ! q, q ! p, m ! r results in
Γ
i
jk ¼ Γ pqr
∂xq
∂xj
∂xr
∂xk
∂xi
∂xp
þ ∂
2
xq
∂xj∂xk
∂xi
∂xp
ð2:4:22Þ
Expression (2.4.22) is the transformation law of the Christoffel symbol of
second kind. The second term to the right of this expression shows that these
coefficients are not the components of tensor. The Christoffel symbols do not
depend only on the coordinate system, but depend also on the rate with which
this coordinate system varies in each point of the space. This variation rate is not
present in the transformation law of tensors.
2.3 Derivatives of the Basis Vectors 87
2.3.9 Linear Transformations
Consider the transformation of coordinates between two coordinate systems given
by linear relation
xj ¼ aji x i þ b j
where a ji and b
j are constants, and with the techniques of successive differentiation
∂xj
∂xi
¼ aji )
∂2xj
∂xi∂xk
¼ 0
then for this kind of transformation of coordinates the Christoffel symbols trans-
form as tensors.
2.3.10 Orthogonal Coordinate Systems
In the orthogonal coordinate systems, the tensorial space EN is defined by the metric
tensor gij 6¼ 0 for i ¼ j and gij ¼ 0 for i 6¼ j.
Putting h2i ¼ gii, where gii does not indicate the summation of the terms, with the
Christoffel symbol of first kind
Γij,k ¼ 1
2
∂gjk
∂xi
þ ∂gik
∂xj
þ ∂gij
∂xk
� �
and with the components of the tensor gij given by
g‘k ¼ 1 ! i ¼ j
0 ! i 6¼ j
(
gij ¼
1
gij
it results for the relation between the Christoffel symbols
Γ ‘ij ¼ g‘kΓij,k ¼
gkkΓij,k ! ‘ ¼ k
0 !! ‘ 6¼ k
(
Varying the indexes:
– i ¼ j ¼ k
88 2 Covariant, Absolute, and Contravariant Derivatives
Γii, i ¼ 1
2
∂gii
∂xj
þ ∂gii
∂xj
� ∂gii
∂xj
� �
¼ 1
2
∂gii
∂xj
¼ hi ∂hi∂xj
Γ kij ¼ Γ iii ¼ giiΓii, i ¼
1
2gii
∂gii
∂xj
¼ 1
2
∂ ‘ngiið Þ
∂xj
¼ ∂ ‘n
ffiffiffiffiffi
gii
p� �
∂xj
¼ 1
hi
∂hi
∂xj
– i ¼ j 6¼k
Γii,k ¼ 1
2
∂gik
∂xi
þ ∂gik
∂xi
� ∂gii
∂xk
� �
as i 6¼ k it implies by definition of the metric tensor that gik ¼ 0, so
Γii,k ¼ �1
2
∂gii
∂xk
¼ �hi ∂hi∂xk
Γ kij ¼ Γ kii ¼ gkkΓii,k ¼ �
1
2gkk
∂gii
∂xk
¼ � hi
hkð Þ2
∂hi
∂xk
– i ¼ k 6¼ j
Γij, i ¼ 1
2
∂gji
∂xi
þ ∂gii
∂xj
þ ∂gij
∂xi
� �
and in an analogous way to the previous case where gji ¼ gij ¼ 0, so
Γij, i ¼ 1
2
∂gii
∂xj
¼ hi ∂hi∂xj
Γ kij ¼ Γ iij ¼ giiΓij, i ¼
1
2gii
∂gii
∂xj
¼ ∂ ‘n
ffiffiffiffiffi
gii
p� �
∂xj
¼ 1
hi
∂hi
∂xj
– for i 6¼ j, j 6¼ k, i 6¼ k it results in Γij,k ¼ 0, for by the definition of the metric
tensor it implies gij ¼ gij ¼ 0 if i 6¼ j, whereby Γ kij ¼ 0.
2.3.11 Contraction
The tensorial expressions at times contain Christoffel symbols. However, the
calculation of their components can be avoided, for an expression can be obtained
that relates the derivative of the natural logarithm of the metric tensor with these
symbols.
2.3 Derivatives of the Basis Vectors 89
Let the Christoffel symbol of second kind
Γ jik ¼
1
2
gmj
∂gkm
∂xi
þ ∂gmi
∂xk
� ∂gik
∂xm
� �
and with contraction of the indexes j and k
Γ jij ¼
1
2
gmj
∂gjm
∂xi
þ ∂gmi
∂xj
� ∂gij
∂xm
� �
The symmetry of the metric tensor provides
gmj
∂gmi
∂xj
¼ gjm ∂gji
∂xm
¼ gmj ∂gij
∂xm
where the second equality was obtained by means of indexes interchanging the
m $ j.
Substituting gmj
∂gmi
∂xj
in the expression of the contracted Christoffel symbol
Γ jij ¼
1
2
gjm
∂gjm
∂xi
The conjugate metric tensor can be written as
gjm ¼ G
jm
g
) g ¼ Gjmgjm
being G jm the cofactor of this matrix and g ¼ detgjm, it follows that
Γ jij ¼
1
2g
Gjm
∂gjm
∂xi
¼ 1
2g
∂g
∂xi
¼ 1
2
∂ ‘ngð Þ
∂xi
¼ ∂ ‘n
ffiffiffi
g
p� �
∂xi
and with the contracted form of the Christoffel symbol of second kind
Γ jij ¼
1ffiffiffi
g
p ∂
ffiffiffi
g
p� �
∂xi
ð2:4:23Þ
that is of great use in manipulations of tensorial expressions, for it reduces the
algebrism in calculating the Christoffel symbol.
For g ¼ jgijj < 0 the analysis is analogous, having only to change the sign of the
determinant in the expression shown in the previous demonstration
Γ iip ¼
∂ ‘n
ffiffiffiffiffiffiffi�gp� �
∂xp
¼ 1ffiffiffiffiffiffiffi�gp ∂
ffiffiffiffiffiffiffi�gp� �
∂xp
ð2:4:24Þ
90 2 Covariant, Absolute, and Contravariant Derivatives
2.3.12 Christoffel Relations
Consider the transformation of the Christoffel symbol of second kind from one
coordinate system Xi to another coordinate systemX
j
, whereby rewriting expression
(2.4.22)
Γ
r
pq ¼ Γmij
∂xr
∂xm
∂xi
∂xp
∂xj
∂xq
þ ∂x
r
∂xj
∂2xj
∂xq∂xp
and multiplying by ∂x
s
∂xr it follows that
∂xs
∂xr
Γ
r
pq ¼ Γmij
∂xs
∂xr
∂xr
∂xm
∂xi
∂xp
∂xj
∂xq
þ ∂x
s
∂xr
∂xr
∂xj
∂2xj
∂xq∂xp
∂xs
∂xr
∂xr
∂xm
¼ δ sm
∂xs
∂xr
∂xr
∂xj
¼ δ sj
∂xs
∂xr
Γ
r
pq ¼ Γmij δ sm
∂xi
∂xp
∂xj
∂xq
þ δ sj
∂2xj
∂xq∂xp
) ∂x
s
∂xr
Γ
r
pq ¼ Γ sij
∂xi
∂xp
∂xj
∂xq
þ ∂
2
xs
∂xq∂xp
∂2xs
∂xq∂xp
¼ ∂x
s
∂xr
Γ
r
pq � Γ sij
∂xi
∂xp
∂xj
∂xq
ð2:4:25Þ
Expression (2.4.25) shows that the second derivative of the coordinate xs can be
decomposed into terms with the first derivatives of this coordinate and the coordi-
nates xi, xj, and with the Christoffel symbols of second kind. This important
expression was deducted in 1869 by Elwin Bruno Christoffel.
Let an inverse transformation for the Christoffel symbol of second kind of the
coordinate system X
j
to another referential system Xi given by
Γ j‘k ¼ Γ
r
pq
∂xp
∂x‘
∂xq
∂xk
∂xj
∂xr
þ ∂x
j
∂xr
∂2xr
∂x‘∂xk
and multiplying both members by ∂x
m
∂xj and proceeding in a manner that is analogous
to the previous one
∂2xm
∂x‘∂xk
¼ ∂x
m
∂xj
Γ j‘k � Γ
m
pq
∂xp
∂x‘
∂xq
∂xk
ð2:4:26Þ
The transformation of the Christoffel symbols from one coordinate system Xi to
another coordinate system X
j
, and from this one to a third coordinates system eXk is
identical to the transformation from Xi directly to eXk, so the transitive property is
valid for the transformations of the Christoffel symbols. This shows that these
symbols form a group.
2.3 Derivatives of the Basis Vectors 91
The Christoffel relation given by expression (2.4.25) can be written as
∂2xk
∂xj∂xm
∂xs
∂xk
¼ Γ sjm �
∂xs
∂xp
∂xk
∂xj
∂xr
∂xm
Γ pkr
and contracting the terms in the indexes s and m
∂2xk
∂xj∂xm
∂xm
∂xk
¼ Γmjm �
∂xm
∂xp
∂xk
∂xj
∂xr
∂xm
Γ pkr
∂2xk
∂xj∂xm
∂xm
∂xk
¼ Γmjm � δ rp
∂xk
∂xj
Γ pkr
Γ
m
jm ¼
∂xk
∂xj
Γ rkr þ
∂2xk
∂xj∂xm
∂xm
∂xk
that is the transformation law of the contracted Christoffel symbol of second kind.
2.3.13 Ricci Identity
Another usual expression in Tensor Calculus is obtained by means of defining the
Christoffel symbol of first kind
Γji,k ¼ 1
2
∂gik
∂xj
þ ∂gjk
∂xi
� ∂gij
∂xk
� �
Γki, j ¼ 1
2
∂gij
∂xk
þ ∂gkj
∂xi
� ∂gik
∂xj
� �
The sum of these two expressions provides the Ricci identity
∂gjk
∂xi
¼ Γji,k þ Γki, j ð2:4:27Þ
In an analogous way, subtracting the second expression from the first expression
of the Christoffel symbol of first kind provides
∂gij
∂xk
� ∂gjk
∂xi
¼ Γkj, i � Γij,k ð2:4:28Þ
Expressions (2.4.27) and (2.4.28) are very useful in manipulations of tensorial
equations.
Exercise 2.3 If Tij and gik are the components of a symmetric tensor and the metric
tensor, respectively, show that TjkΓij,k ¼ 12 Tjk
∂gjk
∂xi .
92 2 Covariant, Absolute, and Contravariant Derivatives
With the Ricci identity
∂gik
∂xi
¼ Γji,k þ Γki, j
1
2
Tjk
∂gjk
∂xi
¼ 1
2
Tjk Γji,k þ Γki, j
� � ¼ 1
2
TjkΓji,k þ TjkΓki, j
� �
Interchanging the indexes j $ k in the last term to the right of the expression and
considering the tensor’s symmetry then
1
2
Tjk
∂gjk
∂xi
¼ 1
2
Tjk Γji,k þ Γki, j
� � ¼ 1
2
TjkΓji,k þ TkjΓji,k
� � ¼ 1
2
� 2TjkΓji,k
1
2
Tjk
∂gjk
∂xi
¼ TjkΓji,k Q:E:D:
2.3.14 Fundamental Relations
The derivative of the metric tensor with respect to an arbitrary variable can be
placed in terms of Christoffel symbols of second kind and the metric tensor, thus
from the definition of this symbol
Γ pik ¼ gpjΓik, j Γ pjk ¼ gpiΓjk, i
Multiplying these two expressions by gpj and gpi, respectively:
gpjΓ
p
ik ¼ gpjgpjΓik, j ) gpkΓ pik ¼ δpjΓik, j
gpiΓ
p
jk ¼ gpigpiΓjk, i ) gpiΓ pjk ¼ δpiΓjk, i
whereby
Γik, j ¼ gpjΓ pik Γjk, i ¼ gipΓ pjk
Adding these two expressions and considering the Ricci identity
∂gik
∂xi
¼ Γji,k þ Γki, j ð2:4:29Þ
and as gip ¼ gpi it results in
∂gij
∂xk
¼ gpjΓ pik þ gipΓ pjk ð2:4:30Þ
2.3 Derivatives of the Basis Vectors 93
With analogous analysis this derivative can be placed in terms of Christoffel
symbols of the second kind and the conjugate metric tensor, and with
gijgkj ¼ δ ik
the derivative is
∂gij
∂xp
gkj þ gij
∂gkj
∂xp
¼ 0 ) ∂g
ij
∂xp
gkj ¼ �gij
∂gkj
∂xp
Multiplying both members of this last expression by gkq:
gkqgkj
∂gij
∂xp
¼ �gkqgij ∂gkj
∂xp
) δqj
∂gij
∂xp
¼ �gkqgij ∂gkj
∂xp
) ∂g
iq
∂xp
¼ �gijgkq ∂gkj
∂xp
and with the Ricci identity
∂gkj
∂xp
¼ Γkp, j þ Γjp,k
that substituted in the previous expression provides
∂giq
∂xp
¼ �gijgkq Γkp, j þ Γjp,k
� � ¼ �gijgkqΓkp, j � gijgkqΓjp,k
For the Christoffel symbol of second kind the result is the following relations
gkqgijΓkp, j ¼ gkqΓ ikp gijgkqΓjp,k ¼ gijΓ qjp
whereby
∂giq
∂xp
¼ �gkqΓ ikp � gijΓ qjp
As Γ qjp ¼ Γ qpj, the result is
∂giq
∂xp
¼ �gijΓ qpj � gkqΓ ikp ð2:4:31Þ
Expressions (2.4.30) and (2.4.31) are well used in the development of tensorial
expressions.
Exercise 2.4 Calculate the Christoffel symbols Γijk and Γkij for the polar coordi-
nates systems, which metric tensor is given by
94 2 Covariant, Absolute, and Contravariant Derivatives
gij ¼ 1 00 x1ð Þ2
 �
The Christoffel symbol of first kind is given by
Γij,k ¼ 1
2
∂gjk
∂xi
þ ∂gik
∂xj
� ∂gij
∂xk
� �
so
g11 ¼ 1 ) g11,1 ¼ 0, g11,2 ¼ 0
g22 ¼ x1ð Þ2 ) g22,1 ¼ 2x1, g22,2 ¼ 0
g12,1 ¼ g12,2 ¼ g21,1 ¼ g21,2 ¼ g22,2 ¼ 0
It follows that
Γ11,1 ¼ Γ11,2 ¼ Γ12,1 ¼ Γ21,1 ¼ Γ22,2 ¼ 0
Γ12,2 ¼ Γ21,2 ¼ x1
Γ22,1 ¼ �x1
In matrix form the result is
Γij, 1 ¼ 0 00 �x1
 �
Γij, 2 ¼ 0 x
1
x1 0
 �
For the Christoffel symbol of second kind it follows thatgij
h i�1
¼ gij ¼
1 0
0
1
x1ð Þ2
24 35
Γ k12 ¼ gk2Γ12,2
Γ k22 ¼ gk1Γ22,1
k ¼ 1 )
Γ112 ¼ g12Γ12,2 ¼ 0
Γ122 ¼ g11 �x1ð Þ ¼ �x1
(
In matrix form the result is
Γ1ij ¼
0 0
0 �x1
 �
Γ2ij ¼
0
1
x1
1
x1
0
264
375
2.3 Derivatives of the Basis Vectors 95
Exercise 2.5 Calculate the Christoffel symbols for the cylindrical coordinates
system, defined by r � x1 , θ � x2 , z � x3, where �1 	 r 	 1, 0 	 θ 	 2π ,
�1 	 z 	 1, which metric tensor and its conjugated tensor are given, respec-
tively, by
gij ¼
1 0 0
0 r2 0
0 0 1
24 35 gij ¼
1 0 0
0
1
r2
0
0 0 1
2664
3775
Using the expressions deducted for the orthogonal coordinate systems:
– i ¼ j ¼ k
Γii, i ¼ 1
2
∂gii
∂xi
Γ11,1 ¼ 1
2
g11,1 ¼ 0 Γ22,2 ¼
1
2
g22,2 ¼ 0 Γ33,3 ¼
1
2
g33,3 ¼ 0
– i ¼ j 6¼ k
Γii,k ¼ �1
2
∂gii
∂xk
Γ11,2 ¼ �1
2
g11,2 ¼ 0 Γ11,3 ¼ �
1
2
g11,3 ¼ 0
Γ22,1 ¼ �1
2
g22,1 ¼ �r Γ22,3 ¼ �
1
2
g22,3 ¼ 0
Γ33,1 ¼ �1
2
g33,1 ¼ 0 Γ33,2 ¼ �
1
2
g33,2 ¼ 0
– i ¼ k 6¼ j
Γij, i ¼ 1
2
∂gii
∂xj
Γ12,1 ¼ 1
2
g11,2 ¼ 0 Γ13,1 ¼
1
2
g11,3 ¼ 0
Γ21,2 ¼ 1
2
g22,1 ¼ r Γ23,2 ¼
1
2
g22,3 ¼ 0
Γ31,3 ¼ 1
2
g33,1 ¼ 0 Γij, i ¼
1
2
g33,2 ¼ 0
– i 6¼ j, j 6¼ k, i 6¼ k all the Christoffel symbols are null.
Putting these symbols in matrix form, the result is
96 2 Covariant, Absolute, and Contravariant Derivatives
Γij, 1 ¼
0 0 0
0 �r 0
0 0 0
24 35 Γij, 2 ¼ 0 r 0r 0 0
0 0 0
24 35 Γij, 3 ¼ 0½ �
For the Christoffel symbols of second kind it follows that
Γ pij ¼ gpkΓij,k
Γ p22 ¼ gp1Γ22,1 Γ p12 ¼ gp2Γ12,2
– p ¼ 1
Γ122 ¼ g11Γ22,1 ¼ r Γ112 ¼ g12Γ12,2 ¼ 0
– p ¼ 2
Γ222 ¼ g21Γ22,1 ¼ 0 Γ212 ¼ g22Γ12,2 ¼
1
r
– p ¼ 3
Γ322 ¼ g31Γ22,1 ¼ 0 Γ313 ¼ g32Γ12,2 ¼ 0
Putting these symbols in matrix form, the result is
Γ1ij ¼
0 0 0
0 �r 0
0 0 0
264
375 Γ2ij ¼
0
1
r
0
1
r
0 0
0 0 0
266664
377775 Γ3ij ¼ 0½ �
Exercise 2.6 Calculate the Christoffel symbols for the spherical coordinates sys-
tem r � x1,φ � x2, θ � x3, where �1 	 r 	 1, 0 	 φ 	 π, 0 	 θ 	 2π, which
metric tensor and its conjugated tensor are given, respectively, by
gij ¼
1 0 0
0 r2 0
0 0 r2 sin 2φ
24 35 gij ¼
1 0 0
0
1
r2
0
0 0
1
r2 sin 2φ
266664
377775
Using the expressions deduced for the orthogonal coordinate systems the result is:
– i ¼ j ¼ k
2.3 Derivatives of the Basis Vectors 97
Γii, i ¼ 1
2
∂gii
∂xi
Γ11,1 ¼ 1
2
g11,1 ¼ 0 Γ22,2 ¼
1
2
g22,2 ¼ 0 Γ33,3 ¼
1
2
g33,3 ¼ 0
– i ¼ j 6¼ k
Γii,k ¼ �1
2
∂gii
∂xk
Γ11,2 ¼ �1
2
g11,2 ¼ 0 Γ11,3 ¼ �
1
2
g11,3 ¼ 0
Γ22,1 ¼ �1
2
g22,1 ¼ �r Γ22,3 ¼ �
1
2
g22,3 ¼ 0
Γ33,1 ¼ �1
2
g33,1 ¼ �r sin 2φ Γ33,2 ¼ �
1
2
g33,2 ¼ �r2 sinφ cosφ
– i ¼ k 6¼ j
Γij, i ¼ 1
2
∂gii
∂xj
Γ12,1 ¼ 1
2
g11,2 ¼ 0 Γ13,1 ¼
1
2
g11,3 ¼ 0
Γ21,2 ¼ 1
2
g22,1 ¼ r Γ23,2 ¼
1
2
g22,3 ¼ 0
Γ31,3 ¼ 1
2
g33,1 ¼ r sin 2φ Γij, i ¼
1
2
g33,2 ¼ r2 sinφ cosφ
– i 6¼ j, j 6¼ k, i 6¼ k all the Christoffel symbols are null.
Putting these symbols in matrix form, the result is
Γij, 1 ¼
0 0 0
0 �r 0
0 0 r sin 2φ
24 35 Γij, 2 ¼ 0 r 0r 0 0
0 0 �r2 sinφ cosφ
24 35
Γij, 3 ¼
0 0 r sin 2φ
0 0 r2 sinφ cosφ
r sin 2φ r2 sinφ cosφ 0
24 35
For the Christoffel symbols of second kind it follows that
Γ pij ¼ gpkΓj,k
– p ¼ k ¼ 1
98 2 Covariant, Absolute, and Contravariant Derivatives
Γ1ij ¼ g11Γij, 1
Γ122 ¼ g11Γ22,1 ¼ �r Γ133 ¼ g11Γ33,1 ¼ �r sin 2φ
– p ¼ k ¼ 2
Γ2ij ¼ g22Γij, 2
Γ212 ¼ g22Γ12,2 ¼
1
r
Γ233 ¼ g22Γ33,2 ¼ � sinφ cosφ
– p ¼ k ¼ 3
Γ3ij ¼ g33Γij, 3
Γ313 ¼ g33Γ13,3 ¼
1
r
Γ323 ¼ g33Γ23,3 ¼ cot φ
Putting these symbols in matrix form, the result is
Γ1ij ¼
0 0 0
0 �r 0
0 0 �r sin 2φ
264
375 Γ2ij ¼
0
1
r
0
1
r
0 0
0 0 sinφ cosφ
266664
377775
Γ3ij ¼
0 0
1
r
0 0 cotφ
1
r
cotφ 0
26664
37775
Exercise 2.7 For the antisymmetric tensor Aijk, show that AijkΓ pij ¼ AijkΓ pjk ¼ Aijk
Γ pik ¼ 0.
The symmetry of the Christoffel symbol of second kind allows writing
AijkΓ pij ¼ AijkΓ pji ¼ 0
and replacing the indexes i ! j it follows that
AijkΓ pij ¼ �AijkΓ pij
the result is
AijkΓ pij ¼ 0
2.3 Derivatives of the Basis Vectors 99
Proceeding in an analogous way for AijkΓpjk and A
ijkΓpik the equalities of what was
enunciated are verified.
Exercise 2.8 Given the expression Γ
i
jk ¼ Γ ijk þ δ ij uk þ δ ikuj, where ui is a covariant
vector and Aij is an antisymmetric tensor, show that AjkΓ
i
jk ¼ 0.
The symmetry of the Christoffel symbol of second kind allows writing
Γ
i
jk ¼ Γ ikj
Γ
i
jk ¼ Γ ikj þ δ ikuj þ δ ij uk
AjkΓ
i
jk ¼ AkjΓ ikj
and with the consideration of the anti-symmetry Ajk ¼ �Akj verifies that
AjkΓ
i
jk ¼ 0 Q:E:D:
2.4 Covariant Derivative
The basic problem treated by the Tensorial Analysis is to research if the derivatives
of tensors generate new tensors, which, in general, does not occur. For the case of
Cartesian coordinate systems the variation rates of the tensors are expressed by
partial derivatives. For instance, the variation rates of a vector’s components
indicate the variation of this vector. However, for the curvilinear coordinate
systems, the expressions for these variation rates are not expressed only by partial
derivatives. Figure 2.3a shows this coordinate systems for the case in which the
vector u has constant modulus and directions (Fig. 2.3a), but their components u1
vary. Figure 2.3b shows the behavior of the vectors u with constant modulus and
different directions, whereby the three vectors are different, but their radial u1 and
tangential u2 ¼ 0 components remain constant. This example indicates the need for
AA
BB
CC u
u
u
uu
u
1X 1X
2X 2X
OO
1u 1u
1u
θ θ
a b
Fig. 2.3 Polar coordinates: (a) vector uwith constant modulus and direction and (b) vector uwith
constant modulus and variable direction
100 2 Covariant, Absolute, and Contravariant Derivatives
researching the variation rates of vectors for the curvilinear coordinate systems,
because the variation rates of their components do not represent the variation of
these vectors.
To exemplify this fact let the scalar function ϕ ¼ �mx, where m is a scalar,
which generates the potential u ¼ �gradϕ, with Cartesian components u1 ¼ m,
u2 ¼ 0. This scalar function in polar coordinates is defined by ϕ ¼ �mr cos θ,
which covariant components of its gradient are given by
∂ϕ
∂r
¼ �m cos θ ∂ϕ
∂θ
¼ mr sin θ
and its physical components areu*1 ¼ ∂ϕ∂r ¼ �m cos θ andu*2 ¼ 1r ∂ϕ∂θ ¼ m sin θ. These
components are not constant. The interpretation of this variation is carried out
admitting a polar coordinates point P(r; θ) being displaced to another point nearby
P
0
r þ dr; θ þ dθð Þ, so the covariant components of the vector u initially given by
u1 ¼ ∂ϕ∂r ¼ �m cos θ u2 ¼
∂ϕ
∂θ
¼ mr sin θ
stay for this new point
δu1 ¼ �m sin θdθ δu2 ¼ m sin θdr þ mr cos θdθ
The elemental variations of these new components are due to the change of
coordinates, and not to the change of vector. This particular indicates the need of
defining a kind of derivative that translates the vector’s variation in an invariant
manner, and leads to the definition of the covariant derivative.
The covariant derivative defines variation rate of parameters that are not depen-
dent on the coordinate systems, and because of that it is of extreme importance in
the expression of physical models, for it generates a new tensor. The denomination
covariant derivative was adopted by Bruno Ricci-Curbastro when conceiving the
Tensor Calculus. The term covariant denotes a kind of partial differentiation of
tensors that generates new tensors with variance one order above the original
tensors. The adjective covariant is used to indicate the tensorial characteristics of
the differentiation of tensors, in which the set of Christoffel symbols Γkij are the
coefficients of connections of the tensorial space EN.
2.4.1 Contravariant Tensor
2.4.1.1 Contravariant Vector
Let the vector u defined by its contravariant components uj:
u ¼ ujgj ð2:5:1Þ
where the unit vectors gj ¼ gj xjð Þ of the curvilinear coordinate system are functions
of the coordinates that define this referential system.
2.4 Covariant Derivative 101
Differentiating the expression (2.5.1)with respect to an arbitrary coordinate xk
results in
∂u
∂xk
¼ ∂u
j
∂xk
gj þ uj
∂gj
∂xk
and using expression (2.4.1)
∂gj
∂xk
¼ Γmjkgm
then
∂u
∂xk
¼ ∂u
j
∂xk
gj þ ujΓmjkgm ð2:5:2Þ
As j is a dummy index in the first term to the right, it can be changed for the
index m:
∂u
∂xk
¼ ∂u
m
∂xk
gm þ ujΓmjkgm ¼
∂um
∂xk
þ ujΓmjk
� �
gm
This expression shows that the covariant derivative of a contravariant vector is
given by the N2 functions
∂ku
m ¼ ∂u
m
∂xk
þ ujΓmjk ð2:5:3Þ
whereby
∂u
∂xk
¼ ∂kumð Þgm ð2:5:4Þ
For the Cartesian systems the Christoffel symbols are null, so the covariant
derivative coincides with the partial derivative ∂u
m
∂xk .
In expression (2.5.3) the result is the variation rate of the vector u along the axes
of the curvilinear coordinate system is given by ∂u
j
∂xk, and the variation of the unit
vectors gj along the axes of this coordinate system is expressed by
∂gj
∂xk. This physical
interpretation of the covariant derivative is associated to the Christoffel symbols Γmjk
that are the connection coefficients of the tensorial space.
Various notations are found in the literature for the term to the left of expression
(2.5.3), the most usual being: ∂kum ¼ Dkum ¼ ∇kum ¼ umjk ¼ umjk ¼ um;k.
102 2 Covariant, Absolute, and Contravariant Derivatives
Expressions (2.5.1) and (2.5.4) are analogous, for∂kum has the aspect of a vector.
The transformation law of contravariant vectors is admitted to demonstrate that
expression (2.5.4) is a tensor, thus
ui ¼ ∂x
i
∂xp
up
which differentiated with respect to the coordinate xj provides
∂ui
∂xj
¼ ∂x
i
∂xp
∂up
∂xq
∂xq
∂xj
� �
þ up ∂
2
xi
∂xq∂xp
∂xq
∂xj
 !
ð2:5:5Þ
and with expression (2.4.26)
∂2xi
∂xq∂xp
¼ Γmpq
∂xi
∂xn
� Γ i‘m
∂x‘
∂xp
∂xm
∂xq
ð2:5:6Þ
Substituting expression (2.5.6) in expression (2.5.5)
∂ui
∂xj
¼ ∂x
i
∂xp
∂up
∂xq
∂xq
∂xj
� �
þ upΓ npq
∂xi
∂xn
∂xq
∂xj
� upΓ i‘m
∂x‘
∂xp
∂xm
∂xq
∂xq
∂xj
∂ui
∂xj
þ upΓ i‘m
∂x‘
∂xp
∂xm
∂xq
∂xq
∂xj
¼ ∂u
p
∂xq
∂xi
∂xp
∂xq
∂xj
þ upΓ npq
∂xi
∂xn
∂xq
∂xj
The dummy index p in the first term to the right can be changed by the index n:
∂ui
∂xj
þ upΓ i‘m
∂x‘
∂xp
∂xm
∂xq
∂xq
∂xj
¼ ∂u
n
∂xq
∂xi
∂xn
∂xq
∂xj
þ upΓ npq
∂xi
∂xn
∂xq
∂xj
and with expression
∂xm
∂xj
¼ δmj
results in
∂ui
∂xj
þ upΓ i‘m
∂x‘
∂xp
δmj ¼
∂un
∂xq
þ upΓ npq
� �
∂xi
∂xn
∂xq
∂xj
With the transformation law of contravariant vectors
u‘ ¼ up ∂x
‘
∂xp
2.4 Covariant Derivative 103
the above expression becomes
∂ui
∂xj
þ u‘Γ i‘j ¼
∂un
∂xq
þ upΓ npq
� �
∂xi
∂xn
∂xq
∂xj
ð2:5:7Þ
It is verified that in expression (2.5.7) the variety in parenthesis transforms as a
mixed second-order tensor, then the covariant derivative of a contravariant vector is
a mixed second-order tensor, i.e., of variance (1, 1).
2.4.2 Contravariant Tensor of the Second-Order
The transformation law of contravariant tensors of the second-order is given by
T
pq ¼ Tij ∂x
p
∂xi
∂xq
∂xj
The derivative of this expression with respect to coordinate x‘ is given by
∂T
pq
∂x‘
¼ ∂T
ij
∂xk
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
þ Tij ∂
2
xp
∂xk∂xi
∂xk
∂x‘
∂xq
∂xj
þ Tij ∂x
p
∂xi
∂2xq
∂xk∂xj
∂xk
∂x‘
and with expression (2.4.26)
∂2xp
∂xk∂xi
¼ Γ rki
∂xp
∂xr
� Γ p‘m
∂x‘
∂xi
∂xm
∂xk
∂2xq
∂xk∂xj
¼ Γ rkj
∂xq
∂xr
� Γ q‘m
∂x‘
∂xj
∂xm
∂xk
then
∂T
pq
∂x‘
¼ ∂T
ij
∂xk
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
þ Tij Γ rki
∂xp
∂xr
� Γ p‘m
∂x‘
∂xi
∂xm
∂xk
� �
∂xk
∂x‘
∂xq
∂xj
þ Tij Γ rkj
∂xq
∂xr
� Γ q‘m
∂x‘
∂xj
∂xm
∂xk
� �
∂xk
∂x‘
∂xp
∂xi
∂T
pq
∂x‘
¼ ∂T
ij
∂xk
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
þ TijΓ rki
∂xp
∂xr
∂xk
∂x‘
∂xq
∂xj
� TijΓ p‘m
∂x‘
∂xi
∂xm
∂xk
∂xk
∂x‘
∂xq
∂xj
þ TijΓ rkj
∂xq
∂xr
∂xk
∂x‘
∂xp
∂xi
� TijΓ q‘m
∂x‘
∂xj
∂xm
∂xk
∂xk
∂x‘
∂xp
∂xi
With
δm‘ ¼
∂xm
∂xk
∂xk
∂x‘
104 2 Covariant, Absolute, and Contravariant Derivatives
it follows that
∂T
pq
∂x‘
¼ ∂T
ij
∂xk
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
þ TijΓ rki
∂xp
∂xr
∂xk
∂x‘
∂xq
∂xj
� TijΓ p‘m
∂x‘
∂xi
∂xq
∂xj
þ TijΓ rkj
∂xq
∂xr
∂xk
∂x‘
∂xp
∂xi
� TijΓ q‘m
∂x‘
∂xj
∂xp
∂xi
In the second term to the right interchanging the indexes i$ r and, likewise, in
the same fourth term with the permutation of the indexes j$ r, it results in
∂T
pq
∂x‘
¼ ∂T
ij
∂xk
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
þ Trj ∂x
p
∂xi
∂xk
∂x‘
∂xq
∂xj
Γ ikr � Tij
∂x‘
∂xi
∂xq
∂xj
Γ
p
‘m
þ Tir ∂x
q
∂xj
∂xk
∂x‘
∂xp
∂xi
Γ jkr � Tij
∂x‘
∂xj
∂xp
∂xi
Γ
q
‘m
and with the expressions
T
‘q ¼ Tij ∂x
‘
∂xi
∂xq
∂xj
T
‘p ¼ Tij ∂x
‘
∂xj
∂xp
∂xi
it follows that
∂T
pq
∂x‘
¼ ∂T
ij
∂xk
þ TrjΓ ikr þ TirΓ jkr
� �
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
� T‘qΓ p‘m � T
‘p
Γ
q
‘m
∂T
pq
∂x‘
þ T‘qΓ p‘m þ T
‘p
Γ
q
‘m ¼
∂Tij
∂xk
þ TrjΓ ikr þ TirΓ jkr
� �
∂xk
∂x‘
∂xp
∂xi
∂xq
∂xj
∂‘T
pq ¼ ∂kTij ∂x
k
∂x‘
∂xp
∂xi
∂xq
∂xj
ð2:5:8Þ
where
∂kT
ij ¼ ∂T
ij
∂xk
þ TrjΓ ikr þ TirΓ jkr ð2:5:9Þ
is the covariant derivative of the contravariant tensor of the second order.
Expression (2.5.8) indicates that the covariant derivative of a contravariant
tensor of the second order is a mixed tensor of the third order, twice contravariant
and once covariant, i.e., variance (2, 1). For the Cartesian coordinates the
Christoffel symbols are null, so the covariant derivative coincides with the partial
derivative ∂T
ij
∂xk .
2.4 Covariant Derivative 105
2.4.2.1 Contravariant Tensor of Order Above Two
To generalize expression (2.5.9) for tensors of order above two, i.e., for instance,
the covariant derivative of the contravariant tensor of the third order, which
expression may be developed by means of the following steps:
(a) The basic structure of its expression is written considering the expression
obtained for the covariant derivative of a contravariant tensor of the second
order
∂pT
ijk ¼ ∂T
ijk
∂xp
þ T


Γ


 þ T


Γ


 þ T


Γ

(b) The indexes of the Christoffel symbols corresponding to the coordinate with
respect to which the differentiation is being carried out are placed
∂pT
ijk ¼ ∂T
ijk
∂xp
þ T


Γ

p þ T


Γ

p þ T


Γ

p
(c) The tensor indexes sequence must be obeyed on placing the contravariant
indexes of the Christoffel symbol
∂pT
ijk ¼ ∂T
ijk
∂xp
þ T


Γ i
p þ T


Γ j
p þ T


Γ k
p
(d) The dummy index q is placed on the Christoffel symbol and in sequential form
in the tensors
∂pT
ijk ¼ ∂T
ijk
∂xp
þ Tq

Γ iqp þ T
q
Γ jqp þ T

qΓ kqp
(e) The remaining indexes are placed in the same sequence in which they appear on
the tensor that is being differentiated
∂pT
ijk ¼ ∂T
ijk
∂xp
þ TqjkΓ iqp þ TiqkΓ jqp þ TijqΓ kqp
This tensor generated by the differentiation of a variance tensor (3, 0) has a
variance (3, 1). Expression (2.5.9) can be generalized by adopting this indexes
placement systematic for a contravariant tensor of order p > 3, then the variance
of this new tensor will always be ( p, 1).
Exercise 2.9 Calculate the covariant derivative of the contravariant components of
vector u expressed in polar coordinates.
In Exercise 2.4 the Christoffel symbols were calculated for the polar coordinates,
given by
106 2 Covariant, Absolute, and Contravariant Derivatives
Γij, 1 ¼ 0 00 �x1
 �
Γij, 2 ¼ 0 x
1
x1 0
 �
Γ1ij ¼
0 0
0 �x1
 �
Γ2ij ¼
0
1
x1
1
x1
0
264
375
The expression for the derivative of the contravariant components of vector u is:
∂ku
m ¼ ∂u
m
∂xk
þ ujΓmjk
– m ¼ 1
∂ku
1 ¼ ∂u
1
∂xk
þ ujΓ1jk
k ¼ 1 ) ∂1u1 ¼ ∂u
1
∂x1
þ ujΓ1j1 ) ∂1u1 ¼
∂u1
∂x1
þ u1Γ111 þ u2Γ121
∂1u
1 ¼ ∂u
1
∂x1
þ 0þ 0 ¼ ∂u
1
∂x1
k ¼ 2 ) ∂2u1 ¼ ∂u
1
∂x2
þ ujΓ1j1 ) ∂2u1 ¼
∂u1
∂x2
þ u1Γ112 þ u2Γ122
∂2u
1 ¼ ∂u
1
∂x2
þ u
1
x1
þ 0 ¼ ∂u
1
∂x2
þ u
1
x1
– m ¼ 2
∂ku
2 ¼ ∂u
2
∂xk
þ ujΓ2jk
k ¼ 1 ) ∂1u2 ¼ ∂u
2
∂x1
þ ujΓ2j1 ) ∂1u2 ¼
∂u2
∂x1
þ u1Γ211 þ u2Γ221
∂1u
2 ¼ ∂u
2
∂x1
þ 0þ u
2
x1
¼ ∂u
2
∂x1
þ u
2
x1
k ¼ 2 ) ∂2u2 ¼ ∂u
2
∂x2
þ ujΓ2j2 ) ∂2u2 ¼
∂u2
∂x2
þ u1Γ212 þ u2Γ222
∂2u
2 ¼ ∂u
2
∂x2
þ u
1
x1
þ 0 ¼ ∂u
2
∂x2
þ u
1
x1
Exercise 2.10 Show that ∂jTij ¼ 1ffiffigp ∂ Tij ffiffigpð Þ∂xj þ TjpΓ ijp.
The expression of the covariant derivative of a contravariant tensor of thesecond
order is given by
2.4 Covariant Derivative 107
∂jT
ij ¼ ∂T
ij
∂xk
þ TmjΓ ikm þ TimΓ jmk
and assuming k ¼ j
∂jT
ij ¼ ∂T
ij
∂xk
þ TmjΓ ijm þ TimΓ jmj
In the study of the contraction of the Christoffel symbol, it was verified that
Γ imj ¼
∂ ‘n
ffiffiffi
g
p� �
∂xr
Substituting this expression in the previous expression
∂jT
ij ¼ ∂T
ij
∂xk
þ TmjΓ ijm þ Tim
∂ ‘n
ffiffiffi
g
p� �
∂xm
As m is a dummy index, it can be changed by the index j in the third term to the
right
∂jT
ij ¼ ∂T
ij
∂xk
þ TmjΓ ijm þ Tij
∂ ‘n
ffiffiffi
g
p� �
∂xj
) ∂jTij ¼ ∂T
ij
∂xk
þ Tij ∂
1
2
‘ng
� �
∂xj
 �
þ TmjΓ ijm
and multiplying and dividing the two terms between brackets by
ffiffiffi
g
p
∂jT
ij ¼ 1ffiffiffi
g
p ffiffiffigp ∂Tij
∂xk
þ Tij 1
2
ffiffiffi
g
p ∂g
∂xj
� �
þ TmjΓ ijm
Changing the indexes j ! p and m ! j in the last term
∂jT
ij ¼ 1ffiffiffi
g
p ffiffiffigp ∂Tij
∂xk
þ Tij 1
2
ffiffiffi
g
p ∂g
∂xj
� �
þ TjpΓ ipj ) ∂jTij ¼
1ffiffiffi
g
p ∂ T
ij ffiffiffigp� �
∂xj
þ TjpΓ ipj
By means of the symmetry of the Christoffel symbol it results
∂jT
ij ¼ 1ffiffiffi
g
p ∂ T
ij ffiffiffigp� �
∂xj
þ TjpΓ ipj Q:E:D:
108 2 Covariant, Absolute, and Contravariant Derivatives
2.4.3 Covariant Tensor
2.4.3.1 Covariant Vector
Let the vector u defined by their covariant components uj:
u ¼ uigj ð2:5:10Þ
where gj ¼ gj xj
� �
are the basis vectors of the curvilinear coordinate system, which
are functions of the coordinates that define this referential system.
Differentiating the expression (2.5.10) with respect to an arbitrary coordinate xk:
∂u
∂xk
¼ ∂ui
∂xk
gi þ ui ∂g
i
∂xk
ð2:5:11Þ
and substituting expression (2.4.4)
∂gi
∂xk
¼ �Γ ikjgj
in expression (2.5.11) the result is
∂u
∂xk
¼ ∂ui
∂xk
gi � uiΓ ikjgj ð2:5:12Þ
As i is a dummy index in the first term to the right of expression (2.5.12), it can
be changed by j:
∂u
∂xk
¼ ∂uj
∂xk
� uiΓ ikj
� �
gj ð2:5:13Þ
thus the covariant derivative of a covariant vector is given by the N2 functions
∂kuj ¼ ∂uj∂xk � u
iΓ ikj ð2:5:14Þ
whereby
∂u
∂xk
¼ ∂kuj
� �
gj ð2:5:15Þ
For the Cartesian coordinate systems the Christoffel symbols are null, so in these
referential systems the covariant derivative of a covariant vector coincides with the
partial derivative
∂uj
∂xk.
2.4 Covariant Derivative 109
Expression (2.5.15) has the aspect of a vector, and to demonstrate that this
expression is a tensor let the transformation law of covariant vectors
up ¼ ∂x
i
∂xp
ui
that differentiated with respect to the coordinate xq provides
∂up
∂xp
¼ ∂ui
∂xk
∂xk
∂xq
∂xi
∂xq
þ ui ∂
2
xi
∂xq∂xp
ð2:5:16Þ
Expression (2.4.25) can be written as
∂2xi
∂xq∂xp
¼ ∂x
i
∂xs
Γ
s
pq � Γ ijk
∂xj
∂xp
∂xk
∂xq
and substituting this expression in expression (2.5.15)
∂up
∂xq
¼ ∂ui
∂xk
∂xk
∂xq
∂xi
∂xp
þ ui ∂x
i
∂xs
Γ
s
pq � Γ ijk
∂xj
∂xp
∂xk
∂xq
� �
∂up
∂xq
� ui ∂x
i
∂xs
Γ
s
pq ¼
∂ui
∂xk
∂xk
∂xq
∂xi
∂xp
� ui ∂x
j
∂xp
∂xk
∂xq
Γ ijk
Replacing the indexes i ! ‘, j ! i in the second term to the right of the
expression, and with
us ¼ ui ∂x
i
∂xs
this expression becomes
∂up
∂xq
� usΓ spq ¼
∂ui
∂xk
� u‘Γ ‘ik
� �
∂xi
∂xp
∂xk
∂xq
Putting
∂qup ¼ ∂up∂xq � usΓ
s
pq
the result is
∂qup ¼ ∂kuið Þ ∂x
i
∂xp
∂xk
∂xq
ð2:5:17Þ
110 2 Covariant, Absolute, and Contravariant Derivatives
Then the covariant derivative of a covariant vector is a covariant tensor of the
second order, i.e., of variance (0, 2).
Various notations are found in the literature for the covariant derivative. For the
covariant vector, the most usual ones are: ∂kum ¼ Dkum ¼ ∇kum ¼ um kj ¼ um;k.
2.4.3.2 Covariant Tensor of the Second Order
The transformation law of covariant tensors of the second order is given by
Tpq ¼ Tij ∂x
i
∂xp
∂xj
∂xq
and differentiating with respect to the coordinate xr
∂Tpq
∂xr
¼ ∂
2
xi
∂xr∂xp
∂xj
∂xq
Tij þ ∂x
i
∂xp
∂2xj
∂xr∂xq
Tij þ ∂x
i
∂xp
∂xj
∂xq
∂Tij
∂xk
∂xk
∂xr
∂2xi
∂xr∂xp
¼ ∂x
i
∂xs
Γ
s
rp � Γ i‘m
∂x‘
∂xp
∂xm
∂xr
∂2xj
∂xr∂xp
¼ ∂x
j
∂xs
Γ
s
rq � Γ j‘m
∂x‘
∂xr
∂xm
∂xq
Substituting these two expressions in the expression of the covariant derivative
∂Tpq
∂xr
¼ ∂x
i
∂xs
Γ
s
rp � Γ i‘m
∂x‘
∂xp
∂xm
∂xr
� �
∂xj
∂xq
Tij þ ∂x
i
∂xp
∂xj
∂xs
Γ
s
rq � Γ j‘m
∂x‘
∂xr
∂xm
∂xq
� �
Tij
þ ∂x
i
∂xp
∂xj
∂xq
∂Tij
∂xk
∂xk
∂xr
∂Tpq
∂xr
¼ Tij ∂x
i
∂xs
∂xj
∂xq
Γ
s
rp � Tij
∂xj
∂xq
∂x‘
∂xp
∂xm
∂xr
Γ i‘m þ Tij
∂xj
∂xs
∂xi
∂xp
Γ
s
rq
� Tij ∂x
i
∂xp
∂x‘
∂xr
∂xm
∂xq
Γ j‘m þ
∂Tij
∂xk
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
In the second term to the right replacing the dummy index m ! k, and
interchanging the indexes i $ ‘, and in the fourth term replacing the indexes
‘ ! k and interchanging the indexes j $ m results in
∂Tpq
∂xr
¼ Tij ∂x
i
∂xs
∂xj
∂xq
Γ
s
rp � T‘j
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
Γ ‘ik þ Tij
∂xi
∂xp
∂xj
∂xs
Γ
s
rq
� Tim ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
Γmkj þ
∂Tij
∂xk
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
2.4 Covariant Derivative 111
and with the transformation law of covariant tensors of the second order
Tsq ¼ Tij ∂x
i
∂xs
∂xj
∂xq
Tps ¼ Tij ∂x
i
∂xp
∂xj
∂xs
∂Tpq
∂xr
� TsqΓ srp � TpsΓ srq ¼
∂Tij
∂xk
� T‘jΓ ‘ik � TimΓmkj
� �
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
Replacing the dummy indexes m ! ‘:
∂Tpq
∂xr
� TsqΓ srp � TpsΓ srq ¼
∂Tij
∂xk
� T‘jΓ ‘ik � Ti‘Γ ‘kj
� �
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
whereby
∂kTpq ¼ ∂kTij
� � ∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
therefore the covariant derivative of a covariant tensor of the second order is a
covariant tensor of the third order, i.e., of variance (0, 3). Whereby the covariant
derivative of a covariant tensor of the second order is given by
∂kTij ¼ ∂Tij∂xk � T‘jΓ
‘
ik � Ti‘Γ ‘kj ð2:5:18Þ
For the Cartesian coordinates the Christoffel symbols are null, so in these
referential systems the covariant derivative of the tensor Tij coincides with the
partial derivative
∂Tij
∂xk .
2.4.3.3 Covariant Tensor of Order Above Two
To generalize expression (2.5.18) for tensors of order above two, i.e., for instance,
the covariant derivative of the covariant tensor of the third order, which expression
may be developed by means of the following steps:
(a) The basic structure of its expression is written considering the expression
obtained for the covariant derivative of a covariant tensor of the second order
∂pTijk ¼ ∂Tijk∂xp þ T


Γ


 þ T


Γ


 þ T


Γ

(b) The indexes of the Christoffel symbols corresponding to the coordinate with
respect to which the differentiation is being carried out are placed
∂pTijk ¼ ∂Tijk∂xp þ T


Γ

p þ T


Γ

p þ T


Γ

p
112 2 Covariant, Absolute, and Contravariant Derivatives
(c) The covariant indexes of the Christoffel symbols must be completed obeying
the sequence of the indexes of the tensor that is being differentiated
∂pTijk ¼ ∂Tijk∂xp þ T


Γ
ip þ T


Γ
jp þ T


Γ
kp
(d) The dummy index q is placed on the Christoffel symbols and in sequential form
in the tensors
∂pTijk ¼ ∂Tijk∂xp þ Tq

Γ
q
ip þ T
q
Γ qjp þ T

qΓ qkp
(e) The remaining indexes are placed in the same sequence in which they appear on
the tensor that is being differentiated
∂pTijk ¼ ∂Tijk∂xp þ TqjkΓ
q
ip þ TiqkΓ qjp þ TijqΓ qkp
This tensor generated by the differentiation of a variance tensor (0, 4). Expres-
sion (2.5.18) can be generalized by adopting this indexes placement systematic for a
covariant tensor of order q > 3, and the variance of this new tensor will always be
0, qþ 1ð Þ.
2.4.4 Mixed Tensor
Consider the transformation law of the mixed tensors of the second
T
m
n ¼ T ij
∂xm
∂xi
∂xj
∂xn
that can be written as
T
m
n
∂xi
∂xm
¼ T ij
∂xj
∂xn
which derivative with respect to coordinate xr is given by
∂T
m
n
∂xr
∂xi
∂xm
þ Tmn
∂2xi
∂xr∂xm
¼ ∂T
i
j
∂xk
∂xk
∂xr
∂xj
∂xn
þ T ij
∂2xj
∂xr∂xn
and with the following expressions
∂2xi
∂xr∂xm
¼ ∂x
i
∂xs
Γ
s
rm � Γ i‘j
∂x‘
∂xm
∂xj
∂xr
∂2xj
∂xr∂xm
¼ ∂x
j
∂xs
Γ
s
mr � Γ j‘p
∂x‘
∂xn
∂xp
∂xr
2.4 Covariant Derivative 113
this expression becomes
∂T
m
n
∂xr
∂xi
∂xm
þ Tmn
∂xi
∂xs
Γ
s
rm � Γ i‘j
∂x‘
∂xm
∂xj
∂xr
� �
¼ ∂T
i
j
∂xk
∂xk∂xr
∂xj
∂xn
þ T ij
∂xj
∂xs
Γ
s
mr � Γ j‘p
∂x‘
∂xn
∂xp
∂xr
� �
As
T
m
n ¼ T pq
∂xm
∂xp
∂xq
∂xn
T ij ¼ T
m
e
∂xi
∂xm
∂xe
∂xj
it follows that
∂T
m
n
∂xr
∂xi
∂xm
þ Tmn
∂xi
∂xs
Γ
s
rm � T pq
∂xm
∂xp
∂xq
∂xn
∂x‘
∂xm
∂xj
∂xr
Γ i‘j
¼ ∂T
i
j
∂xk
∂xk
∂xr
∂xj
∂xn
þ Tme
∂xi
∂xm
∂xe
∂xj
∂xj
∂xs
Γ
s
mr � T ij
∂x‘
∂xn
∂xp
∂xr
Γ j‘p
∂T
m
n
∂xr
∂xi
∂xm
þ Tmn
∂xi
∂xs
Γ
s
rm � T pq δ ‘p
∂xq
∂xn
∂xj
∂xr
Γ i‘j
¼ ∂T
i
j
∂xk
∂xk
∂xr
∂xj
∂xn
þ Tms
∂xi
∂xm
δ es Γ
s
mr � T ij
∂x‘
∂xn
∂xp
∂xr
Γ j‘p
∂T
m
n
∂xr
∂xi
∂xm
þ Tmn
∂xi
∂xs
Γ
s
rm � T ‘q
∂xq
∂xn
∂xj
∂xr
Γ ipj
¼ ∂T
i
j
∂xk
∂xk
∂xr
∂xj
∂xn
þ Tms
∂xi
∂xm
Γ
s
mr � T ij
∂x‘
∂xn
∂xp
∂xr
Γ j‘p
Interchanging the indexes in the second term on the left m $ s, in the last
term on the right, interchanging the indexes j $ ‘ and replacing the indexes p ! k
results in
∂T
m
n
∂xr
∂xi
∂xm
þ T sn
∂xi
∂xm
Γ
m
rs � T ‘q
∂xq
∂xn
∂xj
∂xr
Γ i‘j ¼
∂T ij
∂xk
∂xk
∂xr
∂xj
∂xn
þ Tms
∂xj
∂xm
Γ
s
mr
� T i‘
∂xj
∂xn
∂xk
∂xr
Γ ‘jk
and replacing the indexes j ! k and q ! j in the last term on the left
114 2 Covariant, Absolute, and Contravariant Derivatives
∂T
m
n
∂xr
∂xi
∂xm
þ T sn
∂xi
∂xm
Γ
m
rs � T ‘j
∂xj
∂xn
∂xk
∂xr
Γ i‘k ¼
∂T ij
∂xk
∂xk
∂xr
∂xj
∂xn
þ Tms
∂xj
∂xm
Γ
s
mr � T i‘
∂xj
∂xn
∂xk
∂xr
Γ ‘jk
that can be written as
∂T
m
n
∂xr
þ T snΓmrs � T
m
s Γ
s
mr
� �
∂xi
∂xm
¼ ∂T
i
j
∂xk
þ T ‘j Γ i‘k � T i‘Γ ‘jk
 !
∂xj
∂xn
∂xk
∂xr
then
∂T
m
n
∂xr
þ T snΓmrs � T
m
s Γ
s
mr ¼
∂T ij
∂xk
þ T ‘j Γ i‘k � T i‘Γ ‘jk
 !
∂xm
∂xi
∂xj
∂xn
∂xk
∂xr
ð2:5:19Þ
Putting
∂rT
m
n ¼
∂T
m
n
∂xr
þ T snΓmrs � T
m
s Γ
s
mr ð2:5:20Þ
∂rT
i
j ¼
∂T ij
∂xr
þ T ‘j Γ i‘k � T i‘Γ ‘jk ð2:5:21Þ
the result is the expressions that represent the covariant derivative of the mixed
tensors of the second-order T
m
n and T
i
j, whereby
∂rT
m
n ¼ ∂rT ij
∂xm
∂xi
∂xj
∂xn
∂xk
∂xr
ð2:5:22Þ
Expression (2.5.22) shows that the derivative of a mixed tensor of the second
order is a mixed tensor of the third order, once contravariant and twice covariant,
i.e., of variance (1, 2).
The covariant derivative of a mixed tensor of variance ( p, q) generates a
variance tensor p, qþ 1ð Þ. To generalize expression (2.5.22) for mixed tensors of
order above two, assume as an example the covariant derivatives of a mixed tensor
of the third order of variance (1, 2) and of a mixed tensor of fifth order of variance
(3, 2), which are given, respectively, by the expressions
∂kT
j
p‘ ¼
∂T jp‘
∂xk
� T jq‘Γ qpk � T jpqΓ q‘k þ T qp‘Γ jkq
∂kT
j‘m
rs ¼
∂Tj‘mrs
∂xk
� Tj‘mqs Γ qrk � Tj‘mrq Γ qsk þ Tq‘mrs Γ jkq þ Tjqmrs Γ ‘kq þ Tj‘qrs Γmkq
2.4 Covariant Derivative 115
2.4.5 Covariant Derivative of the Addition, Subtraction,
and Product of Tensors
Expression (2.5.21) shows that the covariant derivative of a mixed tensor comprises
a partial derivative of this tensor and the terms containing Christoffel symbols,
which are always linear in the components of the original tensor. This characteristic
indicates that the covariant differentiation follows the same rules of the ordinary
differentiation of Differential Calculus.
To stress the properties of the covariant derivative let the scalar ϕ(xi) which
ordinary derivative is equal to its covariant derivative, that can be written as the dot
product of the vectors ui and vi expressed in Cartesian coordinates
ϕ xi
� � ¼ uivi
and differentiating
∂kϕ x
i
� � ¼ ∂k uivi� � ¼ d uivið Þ
dxk
¼ du
i
dxk
vi þ ui dvi
dxk
As the covariant and ordinary derivatives are equal, it results in
∂kϕ x
i
� � ¼ ∂k ui� �vi þ ui∂k við Þ
Substituting the expressions of the covariant derivatives of contravariant and
covariant vectors
∂kϕ x
i
� � ¼ ∂ui
∂xk
þ uiΓ ikj
� �
vi þ ui ∂vi∂xk � viΓ
i
kj
� �
¼ ∂u
i
∂xk
vi þ ui ∂vi∂xk
This expression suggests that the covariant derivative of an inner product of
tensors behaves in a manner that is similar to the ordinary derivative. To prove this
assumption, let, for instance, the tensors Aij and Bij for which the following
properties of the covariant derivative are admitted a priori as valid:
(a) ∂k Aij þ Bij
� � ¼ ∂kAij þ ∂kBij;
(b) ∂k Aij � Bij
� � ¼ ∂kAij � ∂kBij;
(c) ∂k AijBij
� � ¼ ∂kAij� �Bij þ Aij ∂kBij� �.
To demonstrate property (a) let the tensor Cij ¼ Aij þ Bij, so
116 2 Covariant, Absolute, and Contravariant Derivatives
∂k Aij þ Bij
� �¼ ∂kCij ¼ ∂Cij∂xk � C‘jΓ ‘ik � Ci‘Γ ‘kj
¼ ∂ Aij þ Bij
� �
∂xk
� A‘j þ B‘j
� �
Γ ‘ik � Ai‘ þ Bi‘ð ÞΓ ‘kj
¼ ∂Aij
∂xk
� A‘jΓ ‘ik � Ai‘Γ ‘kj
� �
þ ∂Bij
∂xk
� B‘jΓ ‘ik � Bi‘Γ ‘kj
¼ ∂kAij þ ∂kBij
In an analogous way, it is possible to prove property (b), replacing only
the addition sign for the subtraction sign in the previous demonstration.
To demonstrate property (c) let the inner product AijB‘m ¼ Cij‘m that generates a
covariant tensor of the fourth order
∂k AijB‘m
� � ¼ ∂kCij‘m ¼ ∂Cij‘m∂xk � Cpj‘mΓ pki � Cip‘mΓ pkj � CijpmΓ pk‘ � Cij‘pΓ pkm
Substituting the expressions of the tensor of the fourth order in terms of the inner
product
∂k AijB‘m
� �¼ ∂ AijB‘m� �
∂xk
� ApjB‘mΓ pki � AipB‘mΓ pk‘ � AijBpmΓ pk‘ � AijB‘pΓ pkm
¼ ∂Aij
∂xk
� ApjΓ pki � AipΓ pkj
� �
B‘m þ Aij ∂B‘m∂xk � BpmΓ
p
k‘ � B‘pΓ pkm
� �
The terms in parenthesis are the covariant derivatives of the covariant tensors of
the second-order, whereby
∂k AijB‘m
� � ¼ ∂kAij� �B‘m þ Aij ∂kB‘mð Þ
thus the covariant derivative of an inner product of tensors follows the same rule as
the derivative of the product of functions in Differential Calculus.
2.4.6 Covariant Derivative of Tensors gij, g
ij, δij
Ricci’s Lemma
The metric tensor behaves as a constant when calculating the covariant
derivative.
2.4 Covariant Derivative 117
The covariant derivative of the metric tensor gij is calculated to demonstrate this
lemma, thus
∂kgij ¼
∂gij
∂xk
� gpjΓ pik � gipΓ pkj
∂kgij ¼
∂gij
∂xk
� Γik, j þ Γkj, i
� �
and with the Ricci identity
∂gij
∂xk
¼ Γik, j þ Γjk, i
and by the symmetry Γjk, i ¼ Γkj, i
∂kgij ¼
∂gij
∂xk
� ∂gij
∂xk
¼ 0
In an analogous way the conjugate metric tensor gij is given by
∂kg
ij ¼ ∂g
ij
∂xk
þ gpjΓ ikp þ gipΓ jkp ð2:5:23Þ
Since
gijg
jp ¼ δpi )
∂ gijg
jp
� �
∂xk
¼ 0 ) ∂gij
∂xk
gjp þ gij
∂gjp
∂xk
¼ 0
and multiplying by giq
giqgjp
∂gij
∂xk
þ giqgij
∂gjp
∂xk
¼ 0 ) giqgjp ∂gij
∂xk
þ δ qj
∂gjp
∂xk
¼ 0 ) ∂g
qp
∂xk
¼ �giqgjp ∂gij
∂xk
it follows that
∂gqp
∂xk
¼ �giqgjp Γik, j þ Γjk, i
� � ¼ �giqgjpΓik, j þ�giqgjpΓjk, i
¼ �giqΓ pik � giqΓ qjk
Replacing the indexes i ! p, q ! i, and p ! j:
∂gqp
∂xk
¼ �gpiΓ jpk � gpjΓ ipk
118 2 Covariant, Absolute, and Contravariant Derivatives
and substituting this expression in expression (2.5.23)
∂kg
ij ¼ �gpiΓ jpk � gpjΓ ipk
� �
þ gpjΓ ikp þ gipΓ jkp
and with the symmetry of gij and the Christoffel symbol of second kind
∂kg
ij ¼ �gipΓ jkp � gpjΓ ikp
� �
þ gpjΓ ikp þ gipΓ jkp ¼ 0
Following the same systematic it implies for the covariant derivative of the
Kronecker delta
∂kδ
i
j ¼
∂δ ij
∂xk
þ δ pj Γ ipk � δ ipΓ pjk ¼ 0þ Γ ijk � Γ ijk ¼ 0
These deductions show that the conjugate metric tensor gij and the Kronecker
delta δij also behave as constants in calculating the covariant derivative.
Exercise 2.11 Show that ∂kT ij ¼ gim ∂kT ij
� �
.
Expressing the mixed tensor by
T ij ¼ gimTmj
the result for its covariant derivative is
∂kT
i
j ¼ ∂kgim
� �
T ij þ gim ∂kT ij
� �
As ∂kgim ¼ 0, it results in
∂kT
i
j ¼ gim ∂kT ij
� �
Q:E:D:
Exercise 2.12 Show that ∂ui∂xj �
∂uj
∂xi
� �
is a covariant tensor of the second order, being
ui a covariant vector.
The covariant derivative of a covariant vector is given by
∂jui ¼ ∂ui∂xj � upΓ
p
ij )
∂ui
∂xj
¼ ∂jui þ upΓ pij
and replacing the indexes i ! j results in
∂uj
∂xi
¼ ∂iuj þ upΓ pji
2.4 Covariant Derivative 119
Carrying out the subtraction presented in the enunciation
∂ui
∂xj
� ∂uj
∂xi
� �
¼ ∂jui þ upΓ pji
� �
� ∂jui þ upΓ pij
� �
and with the symmetry Γ pij ¼ Γ pji
∂ui
∂xj
� ∂uj
∂xi
� �
¼ ∂jui � ∂iuj
As the covariant derivative of a covariant vectoris a tensor of the second order,
then this expression represents a tensor of variance (0, 2).
Exercise 2.13 Show that Γ pij ¼ 12 Tpq ∂Tik∂xj þ
∂Tjk
∂xi �
∂Tij
∂xk
� �
, being Tij a symmetric
tensor and detTij 6¼ 0, and with covariant derivative ∂kTij ¼ 0.
The tensor Tpk can be written under the form
Tpk ¼ gipgjkTij
For the tensor Tij the covariant derivative is given by
∂kTij ¼ ∂Tij∂xk � TpjΓ
p
ik � TipΓ pjk ¼ 0 )
∂Tij
∂xk
¼ TpjΓ pik þ TipΓ pjk
Interchanging the indexes i, j, k cyclically
∂Tjk
∂xi
¼ TpkΓ pji þ TjpΓ pki
∂Tki
∂xj
¼ TpiΓ pkj þ TkpΓ pij
and adding these two expressions and subtracting the one that comes before them,
and considering the tensor’s symmetry
∂Tjk
∂xi
þ ∂Tki
∂xj
� ∂Tij
∂xk
¼ TpkΓ pji þ TjpΓ pki
� �
þ TpiΓ pkj þ TkpΓ pij
� �
� TpjΓ pik þ TipΓ pjk
� �
¼ 2TkpΓ pij
The dummy index p can be changed by the index q, so
∂Tjk
∂xi
þ ∂Tki
∂xj
� ∂Tij
∂xk
¼ 2TkqΓ qij )
1
2
∂Tjk
∂xi
þ ∂Tki
∂xj
� ∂Tij
∂xk
� �
¼ TkqΓ qij
and multiplying by Tpk
1
2
Tpq
∂Tjk
∂xi
þ ∂Tki
∂xj
� ∂Tij
∂xk
� �
¼ TpkTkqΓ qij
120 2 Covariant, Absolute, and Contravariant Derivatives
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and with the contraction
TpkTkq ¼ δpq
it follows that
1
2
Tpq
∂Tjk
∂xi
þ ∂Tki
∂xj
� ∂Tij
∂xk
� �
¼ δpqΓ qij ¼ Γ pij Q:E:D:
2.4.7 Particularities of the Covariant Derivative
To exemplify a particularity of the covariant derivative let the vector u defined by
its covariant components uj ¼ gijui, then
∂kuj ¼ ∂k gijui
� �
¼ ∂kgij
� �
ui þ gij∂kui
and with Ricci’s lemma
∂k giju
i
� �
¼ gij∂kui
The covariant derivative of the contravariant vector is given by
∂ku
i ¼ ∂u
i
∂xk
þ u‘Γ i‘k
so by substitution
∂k giju
i
� �
¼ gij
∂ui
∂xk
þ u‘Γ i‘k
� �
The contravariant components of the vector can be expressed in terms of their
covariant components
∂k giju
i
� �
¼ gij
∂ gi‘u‘
� �
∂xk
þ giju‘Γ i‘k ¼ gij
∂ gi‘u‘
� �
∂xk
þ u‘Γ‘k, j
¼ gij
∂gi‘
∂xk
u‘ þ gijgi‘
∂u‘
∂xk
þ u‘Γ‘k, j
Rewriting expression (2.4.31)
∂gi‘
∂xk
¼ �g‘mΓ imk � gimΓ ‘mk
2.4 Covariant Derivative 121
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which substituted in the previous expression provides
∂k giju
i
� �
¼ gij �g‘mΓ imk � gimΓ ‘mk
� �
u‘ þ gijgi‘
∂u‘
∂xk
þ u‘Γ‘k, j
¼ gij �g‘mu‘Γ imk � gimu‘Γ ‘mk
� �þ δ ‘j ∂u‘∂xk þ u‘Γ‘k, j
¼ �gijumΓ imk � δmj u‘Γ ‘mk þ
∂uj
∂xk
þ u‘Γ‘k, j
¼ umΓmk, j � u‘Γ ‘mk þ
∂uj
∂xk
þ u‘Γ‘k, j
Replacing the dummy indexes ‘ ! m:
∂k giju
i
� �
¼ �umΓmk, j � u‘Γ ‘mk þ
∂uj
∂xk
þ umΓmk, j ) ∂kuj ¼ ∂k gijui
� �
¼ ∂uj
∂xk
� u‘Γ ‘mk
then the covariant derivative of a covariant vector is equal to the covariant deriv-
ative of the product of the metric tensor by the contravariant components of this
vector. This characteristic of the covariant derivative can be generalized for tensors
of order above one, for instance, for a contravariant tensor of the second order the
result is
∂k gipgjqT
pq
� �
¼ ∂kTij
Another particularity of the covariant derivative is its successive differentiation
of a scalar function. Let a scalar function ϕ that represents an invariant, so its
derivative with respect to its coordinate xi is a covariant vector given by
ϕ, i ¼
∂ϕ
∂xi
¼ ∂iϕ
Taking the derivative of this function again, nowwith respect to the coordinate xj:
ϕ, ij ¼
∂2ϕ
∂xj∂xi
¼ ∂j ∂iϕð Þ ¼ ∂
2ϕ
∂xj∂xi
� ∂ϕ
∂xm
Γmij
The dummy index m can be changed, and as the Christoffel symbol is symmet-
ric, it results in
∂j ∂iϕð Þ ¼ ∂i ∂jϕ
� �
Then the covariant derivative of an invariant is commutative.
122 2 Covariant, Absolute, and Contravariant Derivatives
2.5 Covariant Derivative of Relative Tensors
The covariant derivative of relative tensors has characteristics that differ from the
covariant derivative of absolute tensors. For studying the derivatives of these
varieties in a progressive manner, a scalar density of weight W with respect to the
coordinate system X
i
is admitted, given by JWϕ xið Þ. Taking the derivative of this
function
∂ JWϕ
� �
∂xj
¼ JW ∂ϕ
∂xk
∂xk
∂xj
þWJW�1 ∂J
∂xj
ϕ ð2:6:1Þ
The second parcel on the right shows that the gradient of a scalar density is not a
vector. It is verified that for W ¼ 0 the result is a scalar function and
∂ϕ
∂xj
¼ ∂ϕ
∂xk
∂xk
∂xj
is the transformation law of the vectors.
Let the Jacobian cofactor
Cmk ¼
∂xk
∂xm
or
∂xr
∂xj
Cmr ¼ Jδ rr ) Cmr ¼ J
∂xm
∂xk
it follows that
∂J
∂xj
¼ ∂
∂xj
∂xk
∂xm
� �
 �
Cmk )
∂J
∂xj
¼ J ∂
2
xk
∂xj∂xm
∂xm
∂xk
The substitution of this expression in expression (2.6.1) provides
∂ JWϕ
� �
∂xj
¼ JW ∂ϕ
∂xk
∂xk
∂xj
þW ∂
2
xk
∂xj∂xm
∂xm
∂xk
ϕ
 !
ð2:6:2Þ
that is the transformation law of the pseudoscalar JWϕ(xi). Using expression
(2.4.25) the second term in parenthesis can be written as
∂2xk
∂xj∂xm
∂xm
∂xk
¼ Γmj‘ �
∂xm
∂xp
∂xk
∂xj
∂xq
∂x‘
Γ pkq
2.5 Covariant Derivative of Relative Tensors 123
The contraction in the indexes m and ‘ provides
∂2xk
∂xj∂xm
∂xm
∂xk
¼ Γmj‘ �
∂x‘
∂xp
∂xk
∂xj
∂xq
∂x‘
Γ pkq
and with
δqp ¼
∂x‘
∂xp
∂xq
∂x‘
the result is
∂2xk
∂xj∂xm
∂xm
∂xk
¼ Γmj‘ �
∂xk
∂xj
Γ qkq
The substitution of this expression in expression (2.6.2) provides
∂ JWϕ
� �
∂xj
¼ JW ∂ϕ
∂xk
∂xk
∂xj
þWJWΓmj‘ϕ�WJW
∂xk
∂xj
Γ qkqϕ
Let a scalar density which transformation law is given by
ϕ ¼ JWϕ
it results in
∂ JWϕ
� �
∂xj
�WΓmj‘ϕ ¼ JW
∂xk
∂xj
∂ϕ
∂xk
�WΓ qkqϕ
� �
The term in parenthesis to the right represents a covariant pseudovector of
weight W. This expression shows that the covariant derivative of a scalar density
presents an additional term in its expression, in which the factor multiplies the
contracted Christoffel symbol. ForW ¼ 0 this expression is reduced to the gradient
expression of the scalar function ϕ(xi)
∂ϕ
∂xj
¼ ∂ϕ
∂xk
∂xk
∂xj
For a contravariant pseudovector of weight W it follows by means of this
expression that is analogous to the one shown for a scalar density, the next
expression
∂ku
j ¼ ∂u
j
∂xk
þ uqΓ jkq �Wu jΓ qkq ð2:6:3Þ
124 2 Covariant, Absolute, and Contravariant Derivatives
and the contraction of the indexes j and k provides
∂ju
j ¼ ∂u
j
∂xj
þ uqΓ jjq �Wu jΓ qjq
The dummy index j in the third term to the right can be changed by the index q:
∂ju
j ¼ ∂u
j
∂x j
þ 1�Wð ÞuqΓ qjq
If the pseudovector has weightW ¼ 1 this expression is simplified for∂ju j ¼ ∂u j∂x j,
which represents the divergence of vector u j.
The generalization of expression (2.6.3) for a relative tensor of weight W and
variance (1, 1) is given by
∂rT
i
j ¼
∂T ij
∂xr
þ T ‘j Γ j‘k � T i‘Γ ‘jk �WT ijΓ qrq ð2:6:4Þ
For a relative tensor Ti���j��� of weight W and variance ( p, q) it results in
∂rT
i���
j��� ¼
∂Ti���j���
∂xr
þ T ‘j Γ j‘k þ � � �� � �� � �� � �|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
terms relative to
the contravariance
�T i‘Γ ‘jk � � � �� � �� � �� � �|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
terms relative to
the covariance
�WT ijΓ qrq
ð2:6:5Þ
By means of the considerations presented in the first paragraph of item (2.5.4),
and adding that the parcel WTijΓ
q
rq in expression (2.6.5) is linear in terms of the
original tensor, it implies that the rules of ordinary differentiation of Differential
Calculus are applicable to the covariant differentiation of relative tensors.
2.5.1 Covariant Derivative of the Ricci Pseudotensor
The covariant derivative of the Ricci pseudotensor in its contravariant form is
given by
∂iε
jk‘ ¼ ∂i e
jk‘ffiffiffi
g
p
� �
¼
∂ e
ijkffiffi
g
p
� �
∂xi
þ Γ jip
epk‘ffiffiffi
g
p þ Γ kip
ejp‘ffiffiffi
g
p þ Γ ‘ip
ejkpffiffiffi
g
p
∂ e
jk‘ffiffi
g
p
� �
∂xi
¼ 1ffiffiffi
g
p ∂e
jk‘
∂xi
þ ejk‘
∂ 1ffiffigp� �
∂xi
¼ ejk‘
∂ 1ffiffigp� �
∂xi
¼ � e
jk‘
2g
2
3
∂g
∂xi
2.5 Covariant Derivative of Relative Tensors 125
The contraction of the Christoffel symbol provides
∂g
∂xi
¼ 2gΓ ppi
whereby
∂ e
jk‘ffiffi
g
p
� �
∂xi
¼ � e
jk‘ffiffiffi
g
p Γ ppi
Substituting this expression in the expression of covariant derivative
∂iε
jk‘ ¼ ∂i e
jk‘ffiffiffi
g
p
� �
¼ � e
jk‘ffiffiffi
g
p Γ ppi þ Γ jip
epk‘ffiffiffi
g
p þ Γ kipejp‘ffiffiffi
g
p þ Γ ‘ip
ejkpffiffiffi
g
p
The conditions for which this pseudotensor is non-null are that the first three
indexes be different, i.e., j 6¼ k 6¼ ‘, then for j ¼ 1, k ¼ 2, ‘ ¼ 3:
∂iε
jk‘ ¼ ∂i e
jk‘ffiffiffi
g
p
� �
¼ � e
123ffiffiffi
g
p Γ ppi þ Γ jip
ep23ffiffiffi
g
p þ Γ kip
e1p3ffiffiffi
g
p þ Γ ‘ip
e12pffiffiffi
g
p
With p ¼ 1, 2, 3:
∂iε
jk‘ ¼ ∂i e
jk‘ffiffiffi
g
p
� �
¼ � e
123ffiffiffi
g
p Γ11i þ Γ22i þ Γ33i
� �þ Γ ji1 e123ffiffiffigp þ Γ ki2 e123ffiffiffigp þ Γ ‘i3 e123ffiffiffigp
and with the symmetry of the Christoffel symbol it results in
∂iε
jk‘ ¼ ∂i e
jk‘ffiffiffi
g
p
� �
¼ 0
With an analogous expression for the covariant form of the Ricci pseudotensor
εijk ¼ ffiffiffigp eijk it results for its covariant derivative
∂iεijk ¼ ∂i ffiffiffigp eijk� � ¼ ∂ ffiffiffigp eijk� �∂xi � ffiffiffigp epjkΓ pi‘ � ffiffiffigp eipkΓ pj‘ � ffiffiffigp eijpΓ pk‘
The partial derivative referent to the first term to the right is given by
∂
ffiffiffi
g
p
eijk
� �
∂x‘
¼ ∂
ffiffiffi
g
p� �
∂x‘
eijk þ ffiffiffigp ∂ eijk� �∂x‘
but
∂ eijk
� �
∂x‘
¼ 0
126 2 Covariant, Absolute, and Contravariant Derivatives
it results in
∂
ffiffiffi
g
p
eijk
� �
∂x‘
¼ ∂
ffiffiffi
g
p� �
∂x‘
eijk
Expression (2.4.23) can be written as
∂
ffiffiffi
g
p� �
∂x‘
¼ ffiffiffigp Γ pp‘ ) ∂ ffiffiffigp eijk
� �
∂x‘
¼ ffiffiffigp eijkΓ pp‘
Substituting this expression in the expression of the covariant derivative
∂iεijk ¼ ∂i ffiffiffigp eijk� � ¼ ffiffiffigp eijkΓ pp‘ � ffiffiffigp epjkΓ pi‘ � ffiffiffigp eipkΓ pj‘ � ffiffiffigp eijpΓ pk‘
The conditions for which this pseudotensor is non-null are that the first three
indexes be different, i.e., j 6¼ k 6¼ ‘, then for j ¼ 1, k ¼ 2, ‘ ¼ 3:
∂iεijk ¼ ∂i ffiffiffigp eijk� � ¼ ffiffiffigp e123Γ pp‘ � ffiffiffigp ep23Γ p1‘ � ffiffiffigp e1p3Γ p2‘ � ffiffiffigp e12pΓ p3‘
With p ¼ 1, 2, 3:
∂iεijk ¼ ∂i ffiffiffigp eijk� �
¼ ffiffiffigp e123 Γ11‘ þ Γ22‘ þ Γ33‘� �� ffiffiffigp e123Γ11‘ � ffiffiffigp e123Γ22‘ � ffiffiffigp e123Γ33‘
whereby
∂iεijk ¼ ∂i ffiffiffigp eijk� � ¼ 0
These derivatives show that
∂i δ
ijk
pqr
� �
¼ ∂i εijkεpqr
� � ¼ ∂i ε ijk� �εpqr þ εijk∂i εpqr� � ¼ 0
The covariant derivatives of the Ricci pseudotensors εijk, εpqr and the generalized
Kronecker delta δijkpqr being null, it implies that these varieties behave as constants in
the calculation of the covariant derivative.
As an example of an application of this characteristic, let the tensorial expression
εijk∂juk, which covariant derivative is given by
∂i ε
ijk∂juk
� � ¼ εijk∂i ∂juk� �þ ∂juk∂i ∂jε ijk� �
but with
∂iε
ijk ¼ 0
this expression becomes
2.5 Covariant Derivative of Relative Tensors 127
∂i ε
ijk∂juk
� � ¼ εijk∂i ∂juk� �
2.6 Intrinsic or Absolute Derivative
The absolute derivative of a variety is calculated when the coordinates xi vary as a
function of time, i.e., xi ¼ xi tð Þ.
A covariant derivative of an invariant ϕ(xk) is given by
∂kϕ x
k
� � ¼ ∂ϕ xk� �
∂xk
which is equal to its partial derivative.
For the absolute derivative
δϕ xk
� �
δt
¼ ∂ϕ x
k
� �
∂t
þ ∂kϕ xk
� � dxk
dt
¼ ∂ϕ x
k
� �
∂t
þ ∂ϕ x
k
� �
∂xk
dxk
dt
¼ dϕ x
k
� �
dt
then this derivative is equal to its total derivative.
For the vector u(xi) where xi varies as a function of time, which is expressed by
means of its contravariant coordinates, or
u ¼ uk xi tð Þ, t� �gk xi tð Þ� �
The derivative with respect to time is given by
du
dt
¼ d u
kgk
� �
dt
¼ du
k
dt
gk þ uk
∂gk
∂t
dxi
dt
ð2:7:1Þ
and with
duk
dt
¼ ∂u
k
∂t
þ ∂u
k
∂xi
dxi
dt
ð2:7:2Þ
The following expression (item 2.3)
∂gk
∂xi
¼ gmΓmki
substituted in expression (2.7.1) provides
du
dt
¼ du
k
dt
gk þ ukΓmki
dxi
dt
gm
Replacing the indexes k ! m in the first term to the right
128 2 Covariant, Absolute, and Contravariant Derivatives
du
dt
¼ du
m
dt
þ ukΓmki
dxi
dt
� �
gm
thus the absolute derivative of a vector generates a vector.
The covariant derivative of the contravariant vector is written as
δum
δ t
¼ du
m
dt
þ ukΓmki
dxi
dt
ð2:7:3Þ
and substituting expression (2.7.2) in expression (2.7.3)
δum
δ t
¼ ∂u
k
∂t
þ ∂u
k
∂xi
dxi
dt
þ ukΓmki
dxi
dt
) δu
m
δt
¼ ∂u
k
∂t
þ ∂u
k
∂xi
þ ukΓmki
� �
dxi
dt
The covariant derivative of the contravariant vector is given by
∂iu
k ¼ ∂u
k
∂xi
þ ukΓmki
or in vectorial form
∂u
∂xi
¼ ∂iuk
� �
gk
whereby for the absolute derivative of vector u it results that
δum
δ t
¼ ∂u
k
∂t
þ ∂iuk dx
i
dt
or in vectorial form
du
dt
¼ δu
m
δ t
� �
gm
The vector u in terms of their covariant components is given by
u ¼ uk xi tð Þ, t
� �
gk xi tð Þ� �
and with an analogous analysis to the one shown for the contravariant vectors, and
with
gk, i ¼ �Γ kimgm
it results for the absolute derivative of vector u
2.6 Intrinsic or Absolute Derivative 129
δuk
δ t
¼ ∂um
∂t
þ ∂iuk dx
i
dt
where ∂iuk is the covariant derivative of the covariant vector.
These expressions can be generalized for the tensors
δTij
δ t
¼ ∂T
ij
∂t
þ ∂kTij dx
k
dt
ð2:7:4Þ
δTij
δ t
¼ ∂Tij
∂t
þ ∂kTij dx
k
dt
ð2:7:5Þ
δTijm
δ t
¼ ∂T
ij
m
∂t
þ ∂kTijm
dxk
dt
ð2:7:6Þ
The differentiation rules of Differential Calculus are applicable to absolute
differentiation, which can be proven, for instance, for two tensors Aij and Bij,
which algebraic addition generates the tensors Cij ¼ Aij 
 Bij, and which product
results in AijBij. Calculating the absolute derivative of this sum
δCij
δ t
¼ ∂kCij dx
k
dt
¼ ∂k Aij þ Bij
� � dxk
dt
¼ ∂kAij dx
k
dt
þ ∂kBij dx
k
dt
¼ δAij
δ t
þ δBij
δ t
Calculating the absolute derivative of the product of the tensors
δ AijBij
� �
δ t
¼ ∂k AijBij
� � dxk
dt
¼ ∂kAij
� �
Bij
dxk
dt
þ Aij ∂kBij
� � dxk
dt
¼ ∂kAij dx
k
dt
Bij þ Aij∂kBij dx
k
dt
¼ δAij
δ t
Bij þ Aij δBij
δ t
The absolute derivative of vector u calculated along the curve xi ¼ xi tð Þ can be
defined by means of the inner product of its covariant derivative by the tangent
vector to this curve dx
i
dt . For a tensor of order above the unit, and with an analogous
way, the absolute derivative is the inner product of this tensor by the vector tangent
to a curve, then
δTijpqr
δt
¼ ∂kTijpqr
dxk
dt
This definition in conjunction with the considerations made in the first paragraph
of item 2.5.4 indicates that the absolute derivative follows the rules of Differential
Calculus, such as shown for the addition and product of two tensors.
The derivative of the metric tensor gij is given by
130 2 Covariant, Absolute, and Contravariant Derivatives
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δgij
δ t
¼ ∂gij
∂t
þ ∂kgij
dxk
dt
Ricci’s lemma shows that ∂kgij ¼ 0, then
δgij
δ t
¼ ∂gij
∂t
As the metric tensor is independent of time it implies that
∂gij
∂t ¼ 0, whereby it
results that
δgij
δ t ¼ 0, i.e., its absolute derivative is null.
For the tensors gij and δij, which have the same characteristics of the
metric tensor, developing an analysis analogous to the one shown for this tensor
it results in
δgij
δt
¼ ∂g
ij
∂t
þ ∂kgij dx
k
dt
¼ ∂g
ij
∂t
¼ 0
δδ ij
δt
¼ ∂δ
i
j
∂t
þ ∂kδ ij
dxk
dt
¼ ∂δ
i
j
∂t
¼ 0
2.6.1 Uniqueness of the Absolute Derivative
The covariant derivative of a Cartesian tensor coincides with its partial derivative,
then the absolute derivative of this variety, calculated along a curve xi ¼ xi tð Þ, can
be defined by means of the scalar product of this derivative by the vector tangent to
this curve dx
i
dt . For instance, for a Cartesian tensor of variance (2, 3) it results in
δTijpqr
δ t
¼ ∂T
ij
pqr
∂t
dxk
dt
As the partial derivative of a Cartesian tensor is unique, and the scalar product
that defines the absolute derivative generates an invariant, it is possible to conclude
that this derivative is also unique. This analysis can be generalized for arbitrary
tensors.
Exercise 2.14 Calculate the absolute derivative of: (a) giju
ivj; (b) giju
iuj; (c) vector
ui knowing that
δui
δ t ¼ 0.
(a) The expression giju
ivj represents a scalar, and taking the derivative
2.6 Intrinsic or AbsoluteDerivative 131
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δ giju
ivj
� �
δt
¼
d giju
ivj
� �
dt
δ giju
ivj
� �
δ t
¼
δ gij
� �
δ t
uivj þ gij
δ uið Þ
δt
vj þ gijui
δ vjð Þ
δt
¼ gij
δ uið Þ
δt
vj þ gijui
δ vjð Þ
δt
(b) The change of vector v j by vector u j in the expression calculated in the previous
item provides
δ giju
iuj
� �
δt
¼ gij
δ uið Þ
δt
uj þ gijui
δ ujð Þ
δt
Interchanging the indexes i $ j in the first term to the right, and with the
symmetry of the metric tensor results in
δ giju
iuj
� �
δt
¼ gji
δ ujð Þ
δt
ui þ gijui
δ ujð Þ
δt
¼ 2gijui
δ ujð Þ
δt
As giju
iuj ¼ uk k2, it implies that δ giju
iujð Þ
δt ¼ 0, which indicates that δu
j
δt ¼ 0.
(c) The covariant components of the vector are given by
uj ¼ gijui
whereby differentiating
δuj
δ t
¼
δ giju
i
� �
δ t
¼ δgij
δ t
ui þ gij
δui
δ t
¼ gij
δui
δ t
¼ 0
Exercise 2.15 Show that
δ
δ t
dxi
dt
� �
¼ d
2xi
dt2
þ Γ ijk
dxj
dt
dxk
dt
.
Putting
ui ¼ dx
i
dt
results for the absolute derivative of this vector
δui
δt
¼ ∂kui dx
k
dt
¼ ∂u
i
∂xk
þ ujΓ ijk
� �
dxk
dt
¼ ∂u
i
∂xk
dxk
dt
þ ujΓ ijk
dxk
dt
and with
132 2 Covariant, Absolute, and Contravariant Derivatives
uj ¼ dx
j
dt
it implies
δui
δt
¼ ∂u
i
∂xk
dxk
dt
þ Γ ijk
dxj
dt
dxk
dt
It follows that
δ
δt
dxi
dt
� �
¼ d
dt
dxi
dt
� �
þ Γ ijk
dxj
dt
dxk
dt
δ
δt
dxi
dt
� �
¼ d
2xi
dt2
þ Γ ijk
dxj
dt
dxk
dt
Q:E:D:
2.7 Contravariant Derivative
The contravariant derivative is defined considering the tensorial nature of the
covariant derivative, for the raising of the index of tensor ∂k . . . the result is
∂‘ . . . ¼ gk‘∂k . . . ð2:8:1Þ
It is promptly verified with Ricci’s lemma that∂kgij ¼ 0, as well as∂kgij ¼ 0and
∂kδ ij ¼ 0. These relations show that the tensors gij, gij, δij behave as constants in the
calculation of the contravariant derivative.
For the variance tensors ( p, q) the result by means of the expression (2.8.1) is
∂kT������ ¼ gkj∂jT������ ¼ ∂j gkjT������
� � ð2:8:2Þ
Then the contravariant derivative is equivalent to the raising of the indexes of
tensor ∂k . . ., or the covariant derivative of tensor gkjT������.
For instance, for the covariant vector uk:
∂kuk ¼ gkj∂juk ¼ ∂j gkjuk
� � ¼ ∂juj
Problems
2.1 Calculate the Christoffel symbols for the coordinates Xi which metric tensor is
given by
2.7 Contravariant Derivative 133
gij ¼
1 0
0
1
x2ð Þ 2
24 35
Answer:
Γij, 1 ¼ 0 for i, j ¼ 1, 2 Γij, 2 ¼
0 0
0 � 1
x2ð Þ2
24 35
Γ1ij ¼ 0 for i, j ¼ 1, 2 Γ2ij ¼
0 0
0 �1
 �
2.2 Calculate the Christoffel symbols for the coordinate system Xi which metric
tensor and its conjugated metric tensor are given by
gij ¼
1 0 0
0 x1ð Þ 2 0
0 0 x1 sin x2ð Þ 2
264
375 gij ¼
1 0 0
0
1
x1ð Þ 2 0
0 0
1
x1 sin x2ð Þ 2
2666664
3777775
Answer:
Γij, 1 ¼
0 0 0
0 �x1 0
0 0 �x1 sin x2ð Þ 2
264
375 Γij, 2 ¼ 0 x
1 0
x1 0 0
0 0 � x1ð Þ 2 sin x2 cos x2
264
375
Γij, 3 ¼
0 0 x1 sin x2ð Þ 2
0 0 x1ð Þ2 sin x2 cos x2
x1 sin x2ð Þ 2 x1ð Þ 2 sin x2 cos x2 0
2664
3775
Γ1ij ¼
0 0 0
0 �x1 0
0 0 �x1 sin x2ð Þ 2
264
375
Γ2ij ¼
0
1
x1
0
1
x1
0 0
0 0 � x1ð Þ 2 sin x2 cos x2
2666664
3777775 Γ3ij ¼
0 0
1
x1
0 0 cot x2
1
x1
cot x2 0
266664
377775
134 2 Covariant, Absolute, and Contravariant Derivatives
2.3 Calculate the Christoffel symbols of the second kind, where F(x1; x2) is a
function of the coordinates, for the referential system which metric tensor is
gij ¼ 1 00 F x1; x2ð Þ
 �
Answer:
Γ1ij ¼
1 0
0 �1
2
∂F
∂x1
24 35 Γ2ij ¼ 1
1
2F
∂F
∂x1
1
2F
∂F
∂x1
1
2F
∂F
∂x2
2664
3775
2.4 Calculate the Christoffel symbols for the space defined by the metric tensor
gij ¼
�1 0 0 0
0 �1 0 0
0 0 �1 0
0 0 0 e�x
4
2664
3775
Answer: Γ44,4 ¼ �12 e�x
4
; Γ444 ¼ �12.
2.5 Calculate the covariant derivative of the inner product of the tensors Ajk and B
‘m
n
with respect to coordinate xp.
Answer:
∂pA
j
k
� �
B‘mn þ Ajk ∂pB‘mn
� �
2.6 Show that
∂
ffiffi
g
p
gijð Þ
∂xi þ Γ jpq
ffiffiffi
g
p
gpq ¼ 0.
2.7 Contravariant Derivative 135
Chapter 3
Integral Theorems
3.1 Basic Concepts
The integral theorems and the concepts presented in this chapter are treated in
Differential and Integral Calculus of multiple variables.
The approach of this subject is carried in a concise and direct manner, and seeks
solely to provide theoretical subsides so that the gradient, divergence, and curl
differential operators can be physically interpreted.
3.1.1 Smooth Surface
The surface S, open or closed, with upward normal n unique in each point, which
direction is a continuous function of its points, is classified as a smooth surface. For
instance, the surface of a sphere is closed smooth, and the surface of a cube is closed
smooth by parts, for it can be decomposed into six smooth surfaces.
3.1.2 Simply Connected Domain
For every closed curve C defined in the domain D, the region formed by C and its
interior is fully contained in D. This curve defines a region R � D, and D is called
simply connected domain (Fig. 3.1a).
The interior of a circle and the interior of a sphere are simply connected regions.
Two concentric spheres define a simply connected region.
© Springer International Publishing Switzerland 2016
E. de Souza Sánchez Filho, Tensor Calculus for Engineers and Physicists,
DOI 10.1007/978-3-319-31520-1_3
137
3.1.3 Multiply Connected Domain
Multiply Connected Domain is the domain D that contains a region R with
N “holes” (Fig. 3.1b). A circle excluded its center defines a simply connected
domain, and the “hole” is reduced to a point, but the region between two coaxial
cylinders is multiply connected.
3.1.4 Oriented Curve
The closed smooth curve C that limits a region R is counterclockwise oriented if
this region stays to its left, i.e., this curve is positively oriented.
3.1.5 Surface Integral
Consider S a smooth surface by parts with upward unit normal vector n, and ϕ(xi) a
function that represents a smooth curve C over this surface (Fig. 3.2).
Dividing this finite area surface, defined by the function ϕ(xi) in N elementary
areas dSi, i ¼ 1, 2, . . . ,N, where the elementary area contains the point P(xi), and
carrying out the sum
XN
i¼1
ϕ xi
� �
dSi and for N ! 1, thus dSi ! 0, implies the limitðð
S
ϕ xi
� �
dS that represents the integral of surface S.
This limit exists and is independent of the number of divisions made. For a
vectorial function, it results in a similar way
ðð
S
udS.
C
C
D
D
R
1C
2C
R
a bFig. 3.1 Domain:
(a) simply connected and
(b) multiply connected
138 3 Integral Theorems
3.1.6 Flow
Let the vectorial function u dependent on point P(xi) located on the surface S. The
component of u in the direction of the unit normal vector to the surface in this point
is given by the scalar product u � n. With this dot product for all the points located in
the surface elements dS, and carrying out the sum
XN
i¼1
u � ndS, and for N ! 1, and
dS ! 0 implies the integral
F ¼
ðð
S
u � ndS ð3:1:1Þ
that defines the flow of the vectorial function u on the surface S (Fig. 3.3).
The surface area element dS is associated to the area vector dS, with modulus dS
and same direction of n, then
C
n
u
S
dS
( )x iP ( )φ
α
x i
Fig. 3.2 Smooth surface
S
Flow of u
S
dSnu ⋅
dS
a b
Fig. 3.3 Flow: (a) through the surface S and (b) component of the vectorial function u in the
direction normal to the surface S
3.1 Basic Concepts 139
dS ¼ ndS ð3:1:2Þ
Expression (3.1.1) is written as
F ¼
ðð
S
u � ndS ¼
ðð
S
u � dS ð3:1:3Þ
and the integration shown in this expression is independent of the coordinate
system, because the dot product u � n is invariant. In terms of the components of
u, it follows that
F ¼
ðð
S
uinidS ð3:1:4Þ
where ni are the direction cosines of the unit normal vector n.
3.2 Oriented Surface
Let S a surface oriented by means of its upward unit normal vector n, then its outline
C is oriented positively if S stays to its left, thus this curve is anticlockwise oriented.
Figure 3.4 shows a smooth surface S with upwardunit normal vector n, defined in
a Cartesian coordinate system. This surface is expressed by the function z ¼ ϕ x; yð Þ,
which orthogonal projection in plane OX3 determines the region R ¼ S12. The unit
1X
O
dS
ixP
S
2X
2dx
1dx
k
C
12C
12S
1X
O
dS
ixP
S
X
2dx
1dx
k
C
12C
12S
3X
Fig. 3.4 Smooth surface
S with upward unit normal
vector n which outline is a
curve closed smooth C
140 3 Integral Theorems
normal vector n forms an angle α with the axis OX3, being cosα its direction cosine.
The orthogonal projection of the area element dS is given by
dS ¼ dx
1dx2
cos α
The dot product of the unit vectors n and k is given by
n � k ¼ nk k kk k cos α
so
n � kk k ¼ cos α
and therefore
dS ¼ dx
1dx2
n � kk k
Substituting this expression in expression (3.1.3) results in
F ¼
ðð
S
u � ndS ¼
ðð
S
u � n dx
1dx2
n � kk k ð3:1:5Þ
then the surface integral can be calculated as a double integral defined in the
region R. The algebraic value of the flow depends on the field’s orientation. If
α <
π
2
thenF > 0, i.e., the flow “is outward,” and ifα >
π
2
thenF < 0, i.e., “the flow
is inward.”
3.2.1 Volume Integral
Consider the closed smooth surface S that contains a volume V, and ϕ(xi) a function
of position defined on this volume. Dividing V into elementary volumes dVi, then
for the point P(xi) situated over S implies ϕ P xið Þ½ � ¼ ϕ xið Þ. Carrying out the sum of
elementary volumes
XN
i¼1
ϕ xi
� �
dVi and for N ! 1, thus dVi ! 0, results the limitððð
V
ϕ xi
� �
dV that represents the volume integral. This limit exists and is indepen-
dent on the number of divisions. If the function is vectorial, it results in a similar
way
ððð
V
udV.
3.2 Oriented Surface 141
3.3 Green’s Theorem
Consider R a region in the plane OX1X2 involved by the closed smooth curve
C with R to its left. Let the real continuous functions F1(x
1; x2) and F2(x
1; x2),
with continuous partial derivatives in R [ C. Thenðð
R
∂F2
∂x1
� ∂F1
∂x2
� �
dx1dx2 ¼
þ
C
F1dx
1 þ F2dx2
� � ð3:2:1Þ
This theorem is due to George Green (1793–1841) and deals with a generalization
of the fundamental theorem of Integral Calculus for two dimensions.
Figure 3.5 shows the region R involved by the closed smooth curve C, in which
there are lines parallel to the coordinate axes that are tangent to this curve. It is
assumed as a premise that C is intersect by straight lines parallel to the coordinate
axes in a maximum of two points.
The region R is defined by
a 	 x1 	 b, f x1ð Þ 	 x2 	 g x1ð Þ
c 	 x2 	 d, p x2ð Þ 	 x1 	 q x2ð Þ
Let C ¼ AEB [ BFA, with AEB given by x2 ¼ f x1ð Þ, and BFA by x2 ¼ g x1ð Þ. In
an analogous way resultsC ¼ FAE [ EBF, with FAE given by x1 ¼ p x2ð Þ, and EBF
by x1 ¼ q x2ð Þ. With
ðð
R
∂F1
∂x2
dx1dx2 ¼
ðb
a
ðg x1ð Þ
f x1ð Þ
∂F1
∂x2
dx2
2664
3775dx1 ¼ ð
b
a
F1 x
1; x2
� �		g x1ð Þ
f x1ð Þ
dx1 ¼ �
ðb
a
F1 x
1; f x1
� �� �
dx1 �
ðb
a
F1 x
1; g x1
� �� �
dx1
b
d
C
A B
F
E
R
2
X
1
XO
Fig. 3.5 Simply connected
region
142 3 Integral Theorems
The two right members are the line integrals, thenðð
R
∂F1
∂x2
dx1dx2 ¼ �
ð
BFC
F1 x
1; x2
� �
dx1 �
ð
AEB
F1 x
1; x2
� �
dx1 ¼
þ
C
F1 x
1; x2
� �
dx1
If the segment of curve C is parallel to axis OX2, the results of the integrals
are not modified (Fig. 3.6). The integral
ð
F1dx
1 is cancelled in segment GH, for
x1 ¼ constant then dx1 ¼ 0. The same occurs for segment PQ.
With the segmentQGgiven byx2 ¼ f x1ð Þ, and the segmentHPgivenbyx2 ¼ g x1ð Þ:
�
ðð
R
∂F1
∂x2
dx1dx2 ¼
þ
C
F1 x
1; x2
� �
dx1 ð3:2:2Þ
and in the same way
�
ðð
R
∂F2
∂x1
dx1dx2 ¼
þ
C
F2 x
1; x2
� �
dx2 ð3:2:3Þ
Adding expressions (3.2.2) and (3.2.3) results inðð
R
∂F2
∂x1
� ∂F1
∂x2
� �
dx1dx2 ¼
þ
C
F1dx
1 þ F2dx2
� �
Q:E:D:
To prove the validity of this theorem for the more general cases being the region
R ¼ R1 [ R2, in which the integrals are calculated for each subregion (Fig. 3.7).
C
2
X
1XO
P
Q
H
G
Fig. 3.6 Region simply
connects with segments
parallel to one of the
coordinate axes
3.3 Green’s Theorem 143
In the segment ST the line integrals are calculated twice, but as they are of different
direction they cancel each other when they are added, henceþ
TS
F1dx
1 þ F2dx2
� �þ þ
ST
F1dx
1 þ F2dx2
� � ¼ 0
Therefore, the expression of Green’s theorem is valid for the subdivision of
region R (Fig. 3.7). This ascertaining is generalized for a finite region R ¼ R1 [ R2
� � �RN comprising N simple regions, with the outline curvesCi, i ¼ 1, 2, . . . ,N, thenðð
R
∂F2
∂x1
� ∂F1
∂x2
� �
dx1dx2 ¼
XN
i¼1
þ
Ci
F1dx
1 þ F2dx2
� �
The consequence of this division of region R into parts is that this theorem can be
applicable to multiply connected regions (Fig. 3.8). The region involved by the
curve TSBSTAT is simply connected, so Green’s theorem is valid for this region,
hence ðð
R
∂F2
∂x1
� ∂F1
∂x2
� �
dx1dx2 ¼
þ
TSBSTAT
F1dx
1 þ F2dx2
� �
To demonstrate the validity of Green’s theorem for this kind of region, let the
line integrals written in a symbolic wayð
TS
þ
ð
C2
þ
ð
ST
þ
ð
C1
¼
ð
C2
þ
ð
C1
¼
þ
C
C
X 2
X 1O
T
S
R2
R1
Fig. 3.7 Division of the
simply connected region
into two simply connected
regions
144 3 Integral Theorems
for ð
TS
¼ �
ð
ST
therefore ðð
R
∂F2
∂x1
� ∂F1
∂x2
� �
dx1dx2 ¼
þ
C
F1dx
1 þ F2dx2
� �
proves the previous statement.
With the condition
∂F2
∂x1
¼ ∂F1
∂x2
in the region R it follows by Green’s theorem
þ
C
F1dx
1 þ F2dx2
� � ¼ 0
thus the line integral is independent of the path on the closed curveC. To demonstrate
that the admitted condition is necessary and sufficient being the segments C1 and C2
of the curve C shown in Fig. 3.9, for the line integral it follows thatþ
ADBEA
F1dx
1 þ F2dx2
� � ¼ 0
Writing the line integrals of the various segments of curve C under symbolic
form
A
2X
1XO
T
R
1C
2C
1
R S
B
Fig. 3.8 Multiply
connected regions
3.3 Green’s Theorem 145
ð
ADB
þ
ð
BEA
¼ 0
ð
ADB
¼ �
ð
BEA
¼
ð
AEB
then þ
C1
F1dx
1 þ F2dx2
� � ¼ þ
C2
F1dx
1 þ F2dx2
� �
where by
∂F2
∂x1
¼ ∂F1
∂x2
is the necessary and sufficient condition for this
independence.
To admit that a parallel straight line of a coordinated axis intersects the region
R in only two points is not essential, because R can be divided into a number of
subregions which separately fulfill this property.
In vectorial notation with the function F ¼ F1iþ F2 j and the position vector
r ¼ x1iþ x2j, and in differential form dr ¼ dx1iþ dx2j, the line integral along the
curve C is given by þ
C
F1dx
1 þ F2dx2
� � ¼ þ
C
F � dr
A
R
1
C
2
C
B
D
E
Fig. 3.9 Segments C1 and
C2 of the closed curve C
146 3 Integral Theorems
3.4 Stokes’ Theorem
Consider the surface S with upward unit normal vector n involved by a closed
smooth curve C with S to its left, which direction cosines are ni > 0. Let the
continuous real functions F1(x
1; x2; x3), F2(x
1; x2; x3), F3(x
1; x2; x3) with
continuous partial derivatives in S [ C. Thenðð
S
∂F3
∂x2
� ∂F2
∂x3
� �
n1 þ ∂F1∂x3 �
∂F3
∂x1
� �
n2 þ ∂F2∂x1 �
∂F1
∂x2
� �
n3
 �
dS
¼
þ
C
F1dx
1 þ F2dx2 þ F3dx3
� � ð3:3:1Þ
To demonstrate this theorem admit that a line parallel to axis OX3 intersects S only
in a point, then the projection of S on the plane OX1X2 will be the region S12
involved by the closed smooth curve C12 oriented positively (Fig. 3.10), then
dS12 ¼ n3dS ð3:3:2Þ
and n3 > 0.
The equation of surface S is given explicitly by x3 ¼ ϕ x1; x2ð Þ, which allows
substituting the line integral along the curve C by the line integral along curve C12:þ
C
F1 x
1; x2; x3
� �
dx1 ¼
þ
C12
F1 x
1; x2;ϕ x1; x2
� �� �
dx1
In the term to the right the coordinate x2 appears twice, in a direct way and in the
function that represents the surface S. Applying Green’s theorem it follows thatþ
C
F1 x
1; x2;ϕ x1; x2
� �� �
dx1 ¼ �
ðð
S12
∂F1
∂x2
þ ∂F1
∂ϕ
∂ϕ
∂x2
� �
dx1dx2
where
dS12 ¼ dx1dx2
and using expression (3.3.2)
3.4 Stokes’ Theorem 147
þ
C
F1 x
1; x2; x3
� �
dx1 ¼ �
ðð
S
∂F1 x1; x2;ϕx1; x2ð Þ½ �
∂x2
þ ∂F1 x
1; x2;ϕ x1; x2ð Þ½ �
∂ϕ
∂ϕ x1; x2ð Þ
∂x2

 �
n3dS
ð3:3:3Þ
In Integral Calculus of Multiple Variables when studying the surface integrals of
x3 ¼ ϕ x1; x2ð Þ the following expressions are deducted for the direction cosines of
its upward unit normal vector n:
n1 ¼
∂ϕ
∂x1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ ∂ϕ∂x1
� �2
þ ∂ϕ∂x2
� �2r ð3:3:4Þ
1X
O
S
2X
C
12
C
12
S
S
3X
a b
c
1C
2C
S
B
A
Fig. 3.10 Stokes theorem: (a) projection of the smooth surface S with upward unit normal vector
n on the plane OX1X2; (b) surface delimited by the closed smooth curve C; and (c) surface with
outline delimited by more than one curve
148 3 Integral Theorems
n2 ¼
∂ϕ
∂x2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ ∂ϕ∂x1
� �2
þ ∂ϕ∂x2
� �2r ð3:3:5Þ
n3 ¼ 1
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ ∂ϕ∂x1
� �2
þ ∂ϕ∂x2
� �2r ð3:3:6Þ
As the direction cosines are positive, expression (3.3.5) provides
n3
∂ϕ x1; x2ð Þ
∂x2
¼ �n3
and substituting this expression in expression (3.3.3)þ
C
F1 dx
1 ¼ �
ðð
S
n2
∂F1
∂x3
� n3 ∂F1∂x2
� �
dS
In an analogous way for the projections of S on the planes OX2X3 and OX3X1 it
follows that þ
C
F2 dx
2 ¼ �
ðð
S
n3
∂F2
∂x1
� n1 ∂F2∂x3
� �
dS
þ
C
F3 dx
3 ¼ �
ðð
S
n1
∂F3
∂x2
� n2 ∂F3∂x1
� �
dS
Adding these three expressions results inðð
S
∂F3
∂x2
� ∂F2
∂x3
� �
n1 þ ∂F1∂x3 �
∂F3
∂x1
� �
n2 þ ∂F2∂x1 �
∂F1
∂x2
� �
n3
 �
dS ¼
þ
C
F1dx
1 þ F2dx2 þ F3dx3
� �
Q:E:D:
Admit that a line parallel to one of the coordinate axis cuts the surface S only in a
point is not an essential premise. Figure 3.10b, c shows two kinds of surface that do
not fulfill this condition. In this case the surfaces must be divided into a finite
number of subsurfaces, which separately fulfills this hypothesis, allowing Stokes’
theorem to be applied to these subsurfaces, and add the partial results obtained.
Then the line integrals referent to the outlines common to the projections of these
surfaces on a plane of the coordinate system cancel each other, for they are
integrated twice, but with the signs changed.
3.4 Stokes’ Theorem 149
For a surface formed by several closed curves it is also possible to apply Stokes’
theorem. Figure 3.10c shows a surface S limited by the closed and smooth curves C1
and C2. The section S along the curve AB generates a new surface, which outlines
are the curves C1, C2 and AB, considered in opposite directions. Then the line
integral referent to curve AB is calculated twice, but with opposite signs, whereby it
cancels itself, leaving only the results referent to the line integrals of the curves C1
and C2.
The Stokes theorem is a generalization of Green’s theorem for the tridimensional
space. In vectorial notation with the function F ¼ F1iþ F2 jþ F3k and the vector
r ¼ x1iþ x2jþ x3k, which differential is dr ¼ dx1iþ dx2jþ dx3k, the line integral
along the curve C is given byþ
C
F1dx
1 þ F2dx2 þ F3dx3
� � ¼ þ
C
F � dr ð3:3:7Þ
The surface integrals that are present in Stokes’ theorem also have a vectorial
interpretation (item 4.4).
3.5 Gauß–Ostrogradsky Theorem
Consider the volume V with upward unit normal vector n involved by a closed
and smooth surface S, which direction cosines are ni > 0. Let the continuous
real functions F1(x
1; x2; x3), F2(x
1; x2; x3), F3(x
1; x2; x3) with continuous par-
tial derivatives in V [ S. Thenððð
V
∂F1
∂x1
þ ∂F2
∂x2
þ ∂F3
∂x3
� �
dx1dx2dx3 ¼
ðð
S
F1n1 þ F2n2 þ F3n3ð ÞdS
ð3:4:1Þ
Consider a line parallel to axis OX2 that intersects the surface S in a maximum of
two points P and P0, with upward unit normal vector n(P) and n(P0), respectively
(Fig. 3.11). Then the projection of S on OX3X1 will be S31, it follows thatððð
V
∂F2
∂x2
dx1dx2dx3 ¼
ðð
S31
ð
∂F2
∂x2
dx2
� �
dS31 ¼
ðð
S31
F2 P
0
� �
� F2 Pð Þ
h i
dS31
150 3 Integral Theorems
For the area element in this plane and with the direction cosines of the upward
normal n(P) and n(P0):
dS31 ¼ dS Pð Þn2 Pð Þ ¼ �dS P0
� �
n2 P
0
� �
Substituting it results for the point P on S:ððð
V
∂F2
∂x2
dx1dx2dx3 ¼
ðð
S
F2 Pð Þn2 Pð ÞdS
In an analogous way, for the projections of S on the planes OX1X2 and OX2X3:ððð
V
∂F3
∂x3
dx1dx2dx3 ¼
ðð
S
F3 Pð Þn3 Pð ÞdS
ððð
V
∂F1
∂x1
dx1dx2dx3 ¼
ðð
S
F1 Pð Þn1 Pð ÞdS
3
X
O
V
S
Pn
31
dS
C
'
Pn
1
X
2
X
31
S
j
Fig. 3.11 Volume V with upward unit normal vector n, which outline is a closed and smooth
surface S
3.5 Gauß–Ostrogradsky Theorem 151
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The addition of these three expressions resultsððð
V
∂F1
∂x1
þ ∂F2
∂x2
þ ∂F3
∂x3
� �
dx1dx2dx3 ¼
ðð
S
F1n1 þ F2n2 þ F3n3ð ÞdS Q:E:D:
One of the premises adopted in the proof of the theorem of Carl Friedrich Gauß
and Mikhail Vasilievich Ostrogradsky (1801–1861) is that the surface S has two
sides, with a single upward and inward normal in each point. To admit that a
straight line parallel to a coordinate axis intersects the volume V in only two points
is not an essential hypothesis, for V can be divided into a number of subvolumes that
separately fulfill the property admitted initially, allowing the Gauß–Ostrogradsky
theorem to be applied to these subvolumes and adding the partial results obtained.
Figure 3.12a shows the volume V cut in more than two points by a straight line
parallel to axis OX2. The division of V into two volumes V1 and V2, separated by
surface S*, with opposite unit normal vector n1 and n2, being V1 involved by
S1 [ S*, and V2 by S2 [ S*. Then the surface integrals referent to this part common
to the two volumes cancel each other, remaining the integrals on the surfaces S1 and
S2. This makes the applying of this theorem valid to volume V.
If the closed surface S that involves volume V is not smooth, it can be divided
into a finite number of smooth surfaces represented by the functions ϕ(xi), which
have continuous partial derivatives, each one involving a subvolume. This proce-
dure allows applying the Gauß–Ostrogradsky theorem to these subvolumes and
adding the results obtained.
3X
1X
O
2X
2V
1V
2n
1n
V
V
1V
2V
S
S
2S
1S
*
1S
*
2S
*S
*S
a b
Fig. 3.12 Gauß–Ostrogradsky theorem: (a) volume cut in more than two points by a straight line
parallel to a coordinated axis and (b) volume V with voids V1 and V2
152 3 Integral Theorems
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Figure 3.12b shows the volume V involved by the closed surface S with empty
volumes V1 and V2, with which are involved, respectively, by the smooth closed
surfaces S1 and S2. In this case it is necessary to cut the total volume and the
volumes of the voids by a plane π and the surfaces of their outlines to project in this
plane, originating the surfaces S*, S�1 and S
�
2, and then apply the Gauß–Ostrogradsky
theorem considering these surfaces.
In vectorial notation with F ¼ F1iþ F2 jþ F3k resultsððð
V
∂F1
∂x1
þ ∂F2
∂x2
þ ∂F3
∂x3
� �
dx1dx2dx3 ¼
ðð
S
F � ndS ð3:4:2Þ
The volume integral that is present in Gauß–Ostrogradsky theorem also has a
vectorial interpretation (item 4.3).
3.5 Gauß–Ostrogradsky Theorem 153
Chapter 4
Differential Operators
4.1 Scalar, Vectorial, and Tensorial Fields
4.1.1 Initial Notes
The study of the scalar, vectorial, and tensorial fields is strictly related with the
differential operators which are applied to the analytic functions that represent these
fields.
In this chapter the differential operators gradient, divergence, and curl will be
defined, and their physical interpretations, as well as various fundamental relations
with these operators, will be presented. These expressions form the mathematical
backbone for the practical applications of the Field Theory. The conception of fields
is of fundamental importance to the formulation of Tensor Calculus, and allows
definingvarious concepts and deducing several expressions which form the frame-
work for the study of the tensors contained in the tensorial space that defines the
field.
The scalar, vectorial, and tensorial fields are formulations carried out on a point
xi2D, the domain D � EN being an open subset and embedded in the ordinary
geometric space. In these three kinds of fields the formulations are the functions
smooth, continuous, and derivable.
By defining an arbitrary origin in the space EN a biunivocal correspondence is
determined for each domain with a variety, scalar, vector, or tensor that defines the
kind of field.
The scalar and vectorial fields are particular cases of the tensorial fields. The
behavior of a tensorial field is measured by the variation rate of the tensor in the
points contained in the field. In the literature it is usual to call this variation rate as
tensor derivative, which is incorrect, for what exists is the variation rate of the field
defined by this variety, so the proper denomination is variation of the tensorial field.
However, on account of being customary by use, the denomination tensor deriv-
ative will be used in this text to express this variation.
© Springer International Publishing Switzerland 2016
E. de Souza Sánchez Filho, Tensor Calculus for Engineers and Physicists,
DOI 10.1007/978-3-319-31520-1_4
155
4.1.2 Scalar Field
Let a scalar be associated to a point in the Euclidian space E3 given by a function of
the coordinates xi, which is defined asϕ ¼ ϕ xi; tð Þ, i ¼ 1, 2, 3, where t is the time in
the instant in which the scalar is measured.
A scalar field is defined by the function of field ϕ(xi; t), and if the time variable t
is constant, the level surface of the field ϕ xið Þ ¼ C is defined, where C is a constant.
For several values of C there is a family of level surfaces, which characterize the field
geometrically. These surfaces do not intersect, for if they did the function ϕ(xi) would
have to assume various values, which is impossible, for this function has only one
value for each xi.
As an example of scalar field there is a point in the interior of a reservoir
containing liquid, where each particle of this fluid is submitted to a pressure
proportional to the distance of this particle up to the top of the free surface. Another
example is the field of temperatures due to a heat source, where the isotherms are
spherical surfaces, with the temperature decreasing to the points farthest from this
source.
4.1.3 Pseudoscalar Field
If the field function defines a pseudoscalar then the field is pseudoscalar. The
specific mass ρ(xi) of the points of a solid body is an example of this sort of field.
4.1.4 Vectorial Field
If the vector u(xi, t) is associated with the point P(xi) of the space EN, then a
vectorial field is defined, and if t¼ constant the field is homogeneous. For the
space E3 which points are referenced to a Cartesian coordinate system there are
three scalar functions of these points, f 1 x
ið Þ, f 2 xið Þ, f 3 xið Þ, i ¼ 1, 2, 3, which express
the field vector
u xi
� � ¼ f 1 xi� �iþ f 2 xi� �jþ f 3 xi� �k
Field lines are defined for a vectorial field determined by the vectorial function
u(xi), in which for each point P(xi) the field vectors are collinear with the vectors
tangents to these lines (Fig. 4.1). The condition of collinearity between the vector
u(xi) and the tangent vector t(xi) is given by the nullity of cross product
εijkujdxk ¼ 0
156 4 Differential Operators
Developing provides:
– i ¼ 1
ε1jkujdxk ¼ 0 ) ε12ku2dxk ¼ 0 ) ε123u2dx3 ¼ 0 ) u2dx3 ¼ 0
ε13ku3dx2 ¼ 0 ) ε132u3dx2 ¼ 0 ) �u3dx2 ¼ 0
– i ¼ 2
ε2jkujdxk ¼ 0 ) ε213u1dx3 ¼ 0 ) �u1dx3 ¼ 0
ε2jkujdxk ¼ 0 ) ε231u3dx1 ¼ 0 ) u3dx1 ¼ 0
– i ¼ 3
ε3jkujdxk ¼ 0 ) ε312u1dx2 ¼ 0 ) u1dx2 ¼ 0
ε3jkujdxk ¼ 0 ) ε321u2dx1 ¼ 0 ) �u2dx1 ¼ 0
Thus the following system results
u3dx2 ¼ u2dx3
u1dx2 ¼ u2dx1
u3dx1 ¼ u1dx3
8><>: )
u3
u2
¼ dx3
dx2
u1
u2
¼ dx1
dx2
u3
u1
¼ dx3
dx1
8>>>>>><>>>>>>:
Field line s
t
u
P
a b
Fig. 4.1 Vectorial field: (a) field lines and field vector and (b) field vectors
4.1 Scalar, Vectorial, and Tensorial Fields 157
For a flat vectorial field the condition of collinearity between the field vector and
the vector tangent to the field lines is given by
u2dx1 � u1dx2 ¼ 0
The gravitational, the electric, and the magnetic are examples of vectorial fields.
4.1.5 Tensorial Field
The fundamental problem of Tensor Calculus is associated to the concept of
tensorial field. If the tensorial field is fixed the tensor T(xi) is a function of the
coordinates of a point P(xi) situated in the tensorial space EN. When this tensor is
function of xi and other parameters then the tensorial field is variable.
For tensor T xið Þ which components are defined with respect to a curvilinear
coordinates X
i
which origin is the point P(xi), a few difficulties arise in the
calculation of its derivatives, because the local coordinate system varies as a
function of the point. The study of the tensorial fields in a tensorial space EN,
considering curvilinear local coordinate systems, is associated to the basis of this
space.
Exercise 4.1 Calculate the parametric equation of the lines of the vectorial field
u ¼ �x2iþ x1 jþ mk that contains the point of coordinates (1; 0; 0) where m is a
scalar.
The differential equations of the field lines are
dx1
�x2 ¼
dx2
x1
¼ dx3
m
Following with the first two differential relations
x1dx1 þ x2dx2 ¼ 0 )
ð
x1dx1 þ
ð
x2dx2 ¼ C0 ) x1ð Þ2 þ x2ð Þ2 ¼ C1; C1 > 0
and introducing a parameter t
x1 ¼
ffiffiffiffiffiffi
C1
p
cos t x2 ¼
ffiffiffiffiffiffi
C1
p
sin t
so
dx2 ¼
ffiffiffiffiffiffi
C1
p
cos t
� �
dt
and with the differential relations
158 4 Differential Operators
dx2
x1
¼ dx3
m
it follows that ffiffiffiffiffiffi
C1
p
cos t
� �
dtffiffiffiffiffiffi
C1
p
cos t
¼ dx3
m
) dx3 ¼ mdt ) x3 ¼ mtþ C2
As the field line contains the point of coordinates (1; 0; 0), then
1 ¼
ffiffiffiffiffiffi
C1
p
cos t ) t ¼ 2kπ; k ¼ 0, 
 1, . . .
– k ¼ 0 ! C1 ¼ 1;
– k ¼ 0 ! t ¼ 0 so C2 ¼ 0;
– verifying that 0 ¼ mtþ C2 for t ¼ 0.
The parametric equations of the field lines represent a circular helix given by
x1 ¼ cos t; x2 ¼ sin t; x3 ¼ mt
4.1.6 Circulation
Consider the field defined by the vectorial function u and the point P(xi) located on
an open curve C, continuous by parts, smooth, and derivable, which is the
hodograph of the position vector r(s), where s is the curvilinear abscissa, and
admitting that this point varies in the interval a 	 xi 	 b, then the line integral of
this curve is given by
I ¼
ðb
a
u � dr ð4:1:1Þ
where line integral defines the circulation of the vectorial function u on the curve C.
Let u � dr be the differential total of the function ϕ(xi), thus
I ¼
ðb
a
dϕ xi
� � ¼ϕ xi� �		b
a
¼ ϕ bð Þ � ϕ að Þ ð4:1:2Þ
The value of this integral depends only on the extreme points of the interval for
which the function ϕ(xi) is defined, regardless of the integration path. Expression
(4.1.2) is a generalization of the fundamental theorem of the Integral Calculus.
4.1 Scalar, Vectorial, and Tensorial Fields 159
For a closed curve the extreme points of this interval are coincident, which
allows concluding that þ
C
u � dr ¼ 0 ð4:1:3Þ
This expression defines the circulation of vector u along the closed curve C. The
line integral of an open curve C defined in a certain interval will be independent of
the path adopted in this calculation, and will be null if the curve C is closed.
Figure 4.2 shows two types of closed paths of curves defined in domain D—the
single closed path in which there are no self-intersection points and the closed path
with self-intersection points.
For the closed spatial curves defined in the domain D self-intersecting in a finite
number of points, the line integral is calculated dividing the path in a finite number
of single closed paths. For an infinite number of intersections, a reasonable approx-
imation is obtained with the integrals on paths which arepolygonal segments, using
a limit process to achieve a finite number of intersections.
4.2 Gradient
In item 2.2 the gradient of a scalar field was defined, by a function ϕ(xi) which
differential is given by
dϕ ¼ ∂ϕ
∂xi
dxi ð4:2:1Þ
that is called differential parameter of the first order of Beltrami.
Expression (4.2.1) shows that there is no difference between the total differential
dϕ and the absolute differential, which allows adopting the notation ϕ,i for the
partial derivatives of this function. It was also shown that the gradient is a vector
Path
Path
Path
Path
1C 1C
2C
2C
b
b
C
C
D D
a b
Fig. 4.2 Closed curve paths: (a) with no self-intersection and (b) with a finite number of self-
intersections
160 4 Differential Operators
obtained by means of applying a vectorial operator to the scalar function ϕ(xi), that
with respect to a coordinate system Xi is given by
∇� � � ¼ ei ∂� � �
∂xi
For a curvilinear coordinate system X
j
by the chain rule
∂� � �
∂xi
¼ ∂x
j
∂xi
∂� � �
∂xj
and with the transformation law of unit vectors
ei ¼ gk ∂x
i
∂xk
it follows that
∇� � � ¼ ei ∂� � �
∂xi
¼ gk ∂x
i
∂xk
∂� � �
∂xi
¼ gk ∂x
i
∂xk
∂xj
∂xi
∂� � �
∂xj
¼ gkδ jk
∂� � �
∂xj
¼ gk ∂� � �
∂xk
The several notations for the gradient vector are
gradϕ xi
� � ¼ G ϕð Þ ¼ ∇ ϕð Þ ¼ gk ∂ϕ xið Þ
∂xk
¼ gkϕ, k ð4:2:2Þ
This comma notation will hereafter be used in some special case for derivatives
with respect to coordinates. The classic notation for the operator that defines the
gradient of a scalar function is gradϕ(xi), and was introduced by Maxwell, Rie-
mann, and Weber. The other notation is an inverted delta, called nabla operator
(in Greek να
0
βλα ¼ harp), del, atled (inverted delta), expressed as ∇� � � ¼ gk ∂���∂xk.
This notation was designed by Hamilton in 1837, initially was not used to represent
the gradient of a function, but was written with the rotated delta ⊲, and represented
symbolically the Laplace operator ddx
� �2 þ ddy� �2 þ ddz� �2 that was already well used
at the time, thereby the denomination Hamilton operator, or Hamiltonian operator.
Another interpretation for the name nabla is due to Maxwell, who remarks that the
rotated delta calls to cuneiform writing, which name in Hebrew would be this one.
The use of the nabla operator has many advantages with respect to the spelling
grad, especially in the development of expressions, for it reinforces the tensorial
characteristics of the gradient. This symbolic vector enables making the spelling for
the differential operators uniform, and complies with the Vectorial Algebra rules.
Figure 4.3a shows schematically a scalar field defined by a function ϕ(xi), where
in a field line contained in the level surface the ϕ xið Þ ¼ C, being C¼ constant, a
point P is defined, and with an arbitrary origin O for the coordinate system Xi,
4.2 Gradient 161
results in the position vector r, which derivative is the vector dr ¼ dxkgk tangent to
the field line, and denotes the line element. The differential of the scalar function
that represents this field is given by the dot product
dϕ ¼ dr �∇ϕ ¼ dxkgk � g‘
∂ϕ
∂x‘
¼ δ ‘k
∂ϕ
∂x‘
dxk ¼ ∂ϕ
∂xk
dxk
The field represented by the gradient of a function is conservative, thus this
function is called potential, or field gradient.
As the operator nabla is a vector, it is invariant for a change in the coordinate
system, which can be proven admitting ∇ for a coordinate system Xj, and ∇ for a
coordinate system Xi, so by means of the vectors transformation law
∂� � �
∂xk
¼ ∂� � �
∂x‘
� �
∂x‘
∂xk
∇� � � ¼ gk
∂� � �
∂xk
¼ ∂x
k
∂xm
∂x‘
∂xk
gm
∂� � �
∂x‘
¼ δ ‘mgm
∂� � �
∂x‘
¼ g‘ ∂� � �
∂x‘
¼ ∇� � �
therefore the operator ∇ is a vector.
The gradient for the function ϕ ¼ xk, where xk represents a coordinate of the
referential system, is given by ∇ϕ ¼ ∂xk∂xi gi, then the gradient is the unit vector for
the coordinate axis.
For the product of two scalar functions ϕ(xi) and ψ(xi) the result is
P
Cxi =φ
rd
P
td
rd
φ∇
O
b
(P)u
element of curve
line C
a b
Fig. 4.3 Scalar field: (a) gradient and (b) line element
162 4 Differential Operators
grad ϕψð Þ ¼ ∇ ϕψð Þ ¼ gk ∂ ϕψð Þ
∂xk
¼ gk ∂ϕ
∂xk
ψ þ ϕgk ∂ψ
∂xk
¼ ∇ϕð Þψ þ ϕ ∇ψð Þ
In this demonstration it is observed that the nabla operator acts on each parcel of
the expression in a distinct way, maintaining a parcel variable and the other
constant. If it comes before the parcel it acts with a variable, if it comes after, it
acts as a constant.
The gradient can be defined by means of the Gauß-Ostrogradsky theorem. Let
the field be determined by the vectorial function u ¼ vϕ xið Þ, v being a constant
vector, then
∂u1
∂xi
¼ v � ∂ϕ x
ið Þ
∂xi
∂u1
∂x1
þ ∂u
2
∂x2
þ ∂u
3
∂x3
¼ v � ∂ϕ
∂x1
þ ∂ϕ
∂x2
þ ∂ϕ
∂x3
� �
¼ v �∇ϕ
it follows that ððð
V
∂u1
∂x1
þ ∂u
2
∂x2
þ ∂u
3
∂x3
� �
dV ¼
ðð
S
u � ndS
v �
ððð
V
∇ϕdV ¼ v �
ðð
S
u � ndS
v �
ððð
V
∇ϕdV �
ðð
S
u � n dS
0@ 1A ¼ 0
For the point P in the scalar field ϕ(xi) contained in an elementary volume, and
with the component ∂ϕ∂x1 of ∇ϕ by the mean value theorem of the Integral Calculusððð
V
∂ϕ
∂x1
dV ¼ ∂ϕ
∂x1
� �
P*
V
where P* is the midpoint of volume V.
Applying the Gauß-Ostrogradsky theorem
∂ϕ
∂x1
� �
P*
¼ 1
V
ðð
S
ϕn1dS
where n1 is the direction cosine of the angle between the upward normal unit vector
n and the coordinate axis OX1.
4.2 Gradient 163
When the point P approaches the point P*, the volume V and the surface S also
come close to P, and with continuous function ϕ(xi) and its partial derivatives it
results in
∂ϕ
∂x1
� �
P
¼ lim
V!0
1
V
ðð
S
ϕn1dS
For the coordinates x2 and x3 the result with analogous formulations is
∂ϕ
∂x2
� �
P
¼ lim
V!0
1
V
ðð
S
ϕn2dS
∂ϕ
∂x3
� �
P
¼ lim
V!0
1
V
ðð
S
ϕn3dS
If these limits exist, the gradient of the scalar function ϕ(xi) in point P is
determined by
∇ϕ xi
� � ¼ lim
V!0
1
V
ðð
S
ϕ xi
� �
ndS ð4:2:3Þ
that is valid for any coordinate system, which shows that the gradient is independent
of the coordinate system.
4.2.1 Norm of the Gradient
The norm of the gradient is given by
∇ϕk k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
∇ϕ �∇ϕk k
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gij
∂ϕ
∂xi
∂ϕ
∂xj
r
ð4:2:4Þ
A few authors use the spelling Δ1ϕ ¼ gij ∂ϕ∂xi ∂ϕ∂xj to designate the first differential
parameter of Beltrami.
For the orthogonal coordinate systems gij ¼ δij:
∇ϕk k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
∂ϕ
∂xi
� �2s
ð4:2:5Þ
164 4 Differential Operators
4.2.2 Orthogonal Coordinate Systems
Consider the point P(xi) be coincident with the origin of the curvilinear orthogonal
coordinate X
j
, and r its position vector with respect to the Cartesian coordinate Xj.
Rewriting expression (2.3.4) the result for this vector’s differential is
dr ¼ ∂r
∂xi
dxi
with
∂r
∂xi
¼ higi
then
hi ¼ ∂r
∂xi
���� ����
where gi are the unit vectors of the coordinate system X
j
, and hi ¼ ffiffiffiffiffiffiffiffig iið Þp are the
metric tensor coefficients that represent scale factors of the magnitudes of the
vectors tangents to the curves of this coordinate system, where the indexes in
parenthesis do not indicate summation. They are called Lamé coefficients for
orthogonal coordinate systems.
The differential of the scalar function ϕ(xi) is given by
dϕ ¼ ∂ϕ
∂xi
dxi
but
dϕ ¼ ∇ϕ � dr
dr ¼ hidxigi
then
dϕ ¼ ∂ϕ
∂xi
dxi
� �
gi � dr
whereby
∇ϕ ¼ 1
hi
∂ϕ
∂xi
� �
gi ð4:2:6Þ
4.2 Gradient 165
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that provides the components of the gradient vector in a curvilinear orthogonal
coordinate system. The physical components of the vector ∇ϕ are given by
∇ϕð Þ* ¼ 1
hi
∂ϕ
∂xi
ð4:2:7Þ
4.2.3 Directional Derivative of the Gradient
Figure 4.3b shows the differential element of the line contained in the level surface
ϕ xið Þ ¼ C and its tangent unit vector t, collinear with vector dr, which allows
writing for the line element dr ¼ tds.
The geometric interpretationof the gradient of a scalar field is given by the dot
product
dϕ
ds
¼ t �∇ϕ ð4:2:8Þ
that defines the field directional derivative. The symbol∇ϕ characterizes the field,
and unit vector t being independent of the function ϕ(xi), this indicates the direction
in which the derivative is calculated. If∇ϕ xið Þ exists in the point P, defined by the
field ϕ(xi), it will be possible to calculate the directional derivative of this function
in all the directions of the field. Then the field ϕ(xi) is non-homogeneous.
Let α be the angle between the two vectors from expression (4.2.8), the dot
product provides
dϕ
ds
¼ tk k ∇ϕk k cos α ¼ ∇ϕk k cos α
As ϕ¼ constant, it results in dϕ ¼ 0, so dr �∇ϕ ¼ 0, then the vector ∇ϕ is
perpendicular to the vector dr.
The variation rate of the field defined by the function ϕ is maximum in the
direction of ∇ϕ, for α ¼ 0 results in cos α ¼ 1, then
dϕ
dn
� �
max
¼ ∇ϕk k > 0
The directional derivative is calculated in the direction of the unit normal vector
n to the level surface ϕ xið Þ ¼ C (Fig. 4.4), thus
∇ϕ ¼ dϕ
dn
n ð4:2:9Þ
166 4 Differential Operators
4.2.4 Dyadic Product
The nabla operator applied to a vectorial function u ¼ ukgk results in the dyadic
product
∇� u ¼ ∇u ¼ T
T ¼ gi ∂u
∂xi
¼ gi ∂ u
kgk
� �
∂xi
¼ gi ∂u
k
∂xi
gk þ uk
∂gk
∂xi
� �
and with expression (2.3.10)
∂gk
∂xi
¼ Γmkigm
it follows that
T ¼ gi ∂u
k
∂xi
gk þ ukΓmkigm
� �
Interchanging the indexes m $ k in the second member to the right
T ¼ ∂u
k
∂xi
þ umΓ kmi
� �
gi � gk
The covariant derivative of a contravariant vector results in a variance tensor
(1, 1)
tP
Cxi =φ
Fig. 4.4 Interpretation of
the gradient as a vector
normal to surface
4.2 Gradient 167
http://dx.doi.org/10.1007/978-3-319-31520-1_2
T ki ¼ ∂iuk ¼
∂uk
∂xi
þ umΓ kmi
then
T ¼ T ki gi � gk ð4:2:10Þ
This analysis shows that the gradient and the covariant derivative represent a
same concept, i.e., they represent the derivative of a scalar, vectorial, or tensorial
function, increasing their variance from one unit.
Formulating an analogous analysis for a covariant vector
T ¼ ∇� u ¼ ∇u ¼ gi ∂u
∂xi
T¼ gi ∂ ukg
k
� �
∂xi
¼ gi ∂uk
∂xi
gk þ uk ∂g
k
∂xi
� �
¼ gi ∂uk
∂xi
gk � umΓmkigm
� �
¼ gi ∂uk
∂xi
� ukΓ kmi
� �
gi � gk
then
∇� u ¼ ∇u ¼ ∂iukð Þgi � gk ¼ Tikgi � gk ð4:2:11Þ
The differential of a vectorial field is a vector, for the differential of vector u:
du ¼ dr �∇� u ¼ dxigi
� �
∂iukg
i � gk� � ¼ dxi∂iukgi � gi � gk
whereby
du ¼ dxi∂iukgk ð4:2:12Þ
For the fields vectorial there is the directional derivative
du
ds
¼ t �∇� u ð4:2:13Þ
The same considerations formulated for the directional derivative of a scalar
field ϕ(xi) are applicable to the vectorial fields. The physical components for the
gradient of vector u* are obtained considering the physical components of the
second-order tensor.
168 4 Differential Operators
4.2.5 Gradient of a Second-Order Tensor
The generalization of the concept of gradient for an arbitrary tensorial field is
immediate. For a coordinate system Xi with unit vector g‘, and with the tensor T
defining the tensorial field
∇� T ¼ gradT ¼ g‘ ∂T
∂x‘
ð4:2:14Þ
Then the gradient of a tensor T is calculated by nabla operator ∇ ¼ g‘ ∂���∂x‘
applying to this tensor. This operator is defined for a contravariant base.
The tensorial product of the nabla operator by the second-order tensor T is
given by
∇� T ¼ ∇T ¼ gm ∂ T
kigk � gi
� �
∂xm
¼ gm ∂T
ki
∂xm
gk � gi þ Tki
∂gk
∂xm
� gi þ Tkigk
∂gi
∂xm
� �
and with the expressions
∂gk
∂xm
¼ Γ pkmgp
∂gi
∂xm
¼ Γ pimgp
it follows that
∇� T ¼ gm ∂T
ki
∂xm
gk � gi þ TkiΓ pkmgp � gi þ TkiΓ pimgk � gp
� �
Interchanging the indexes p $ k in the second term to the right, and the indexes
p $ i in the third term
∇� T ¼ ∂T
ki
∂xm
þ TkiΓ pkm þ TkiΓ pim
� �
gm � gk � gi
this expression becomes
∇� T ¼ ∂mTkigm � gk � gp ð4:2:15Þ
and shows that the gradient of a second-order tensor is a variance tensor (1, 2). The
other components of the gradient of tensor T are given by expressions (2.5.18) and
(2.5.21).
The generalization of the definition of the gradient of a third-order tensor T is
immediate. The components of the fourth-order tensor that result from applying this
operator to tensor T being given by their covariant derivatives, for instance, for
tensor Tijk:
4.2 Gradient 169
http://dx.doi.org/10.1007/978-3-319-31520-1_2
http://dx.doi.org/10.1007/978-3-319-31520-1_2
∇� T ¼ ∂Tijk
∂x‘
� TmjkΓmi‘ � TimkΓmj‘ � TijmΓmk‘
� �
g‘ � gi � g j � gk
For a tensor T of order p the variety ∇� T is a tensor of order pþ 1ð Þ.
The differential of a tensorial field is a tensor, is gives by then the differential of
the second-order tensor T thus:
dT¼ dr �∇� T ¼ dxjgj � ∂mTkigm � gk � gp ¼ dxj∂mTkigj � gm � gk � gp
¼ dxj∂mTkiδmj gk � gp
whereby
dT ¼ dxj∂jTkjgk � gp ð4:2:16Þ
The physical components for the gradient of tensor T* are obtained considering
the physical components of the tensor of the third order.
The same considerations formulated for the directional derivative of a scalar
field ϕ(xi) and of a vectorial field are applicable to the tensorial fields, where
dT
ds
¼ t �∇� T ð4:2:17Þ
4.2.6 Gradient Properties
The ascertaining achieved in the previous paragraphs allow establishing the condi-
tions so that a vector is gradient of a scalar function, for if the vector u(xi) defined in
a single or multiply connected region, and if the line integral
ð
C
u � dr is independent
of the path, then a scalar function ϕ(xi) exists and fulfills the condition u ¼ ∇ϕ xið Þ
in all of this region of the space.
The gradient operator applied to the addition of two tensors provides
∇� Tþ Að Þ ¼ g‘ ∂ Tþ Að Þ
∂x‘
¼ g‘ ∂T
∂x‘
þ g‘ ∂A
∂x‘
¼ ∇� Tþ∇� A
The applying of this operator to the multiplication of the scalar m by the tensor T
provides
∇� mTð Þ ¼ g‘ ∂ mTð Þ
∂x‘
¼ mg‘ ∂T
∂x‘
¼ m∇� T
These two demonstrations prove that the gradient is a linear operator, which is
already implicit, because it is a vector.
170 4 Differential Operators
Exercise 4.2 Calculate: (a) v �∇u; (b) ∇ u � vð Þ.
(a) The gradient for the field defined by a vectorial function is given by
∇u ¼ ∂iukgi � gk
With
v ¼ v‘g‘
it follows that
v �∇u ¼ v‘g‘ � ∂iukgi � gk ¼ v‘∂iukg‘ � gi � gk ¼ v‘∂iukδ i‘gk ¼ vi ∂iukð Þgk
Thus for the Cartesian coordinates
v �∇u ¼ vi ∂ukð Þ∂xi gk
(b) The gradient of the scalar field represented by the dot product of the vectorial
functions u and v is given by
∇ u � vð Þ ¼ gk ∂ u
ivið Þ
∂xk
¼ ∂kui
� �
vi þ ui ∂kvið Þ
� �
gk
and with the expressions
∂ku
i ¼ ∂u
i
∂xk
þ umΓ imk ∂kvi ¼
∂vi
∂xk
� vmΓmik
it follows that
∇ u � vð Þ ¼ ∂u
i
∂xk
vi þ umΓ imkvi þ ui
∂vi
∂xk
� uivmΓmik
� �
gk
In the last term in parenthesis interchanging the indexes i $ m:
∇ u � vð Þ ¼ ∂u
i
∂xk
vi þ umΓ imkvi þ ui
∂vi
∂xk
� umviΓ imk
� �
gk
then
∇ u � vð Þ ¼ ∂u
i
∂xk
vi þ ui ∂vi∂xk
� �
gk
4.2 Gradient 171
For the Cartesian coordinates
∇ u � vð Þ ¼ ∂ui
∂xk
vi þ ui ∂vi∂xk
� �
gk
Another way of expressing ∇ u � vð Þ is to use the expression between the
covariant derivative of a covariant vector and the covariant derivative of a
contravariant vector, which is given by
∂kum ¼ gim∂kui
The multiplying of this expression by gin provides
gin∂kum ¼ gingim∂kui ¼ δnm∂kui ) ∂kui ¼ gim∂kumm
whereby
vi∂ku
i ¼ vigim∂kum ¼ vm∂kum
Replacing the dummy index m ! i:
vi∂ku
i ¼ vi∂kui
and by substitution
∇ u � vð Þ ¼ vi∂kui þ ui∂kvi
� �
gk
Adding and subtracting the terms vi∂iuk and ui∂ivk
∇ u � vð Þ ¼ vi ∂kui � ∂iukð Þ þ vi∂iuk þ ui ∂kvi � ∂ivkð Þ þ ui∂ivk
� �
gk
and with the expressions
v� ∇� uð Þ ¼ vi ∂kui � ∂iukð Þgk u� ∇� vð Þ ¼ ui ∂kvi � ∂ivkð Þgk
v �∇u ¼ vi ∂iukð Þgk u �∇v ¼ ui ∂ivkð Þgk
it results in
∇ u � vð Þ ¼ v �∇uþ v� ∇� uð Þ þ u �∇vþ u� ∇� vð Þ
For the particular case in which u ¼ v:
v �∇v ¼ 1
2
∇v2 � v� ∇� vð Þ
172 4 Differential Operators
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Exercise 4.3 Calculate the gradient of the scalar function ϕ(xi) expressed in
cylindrical coordinates.
Forthe cylindrical coordinates
ffiffiffiffiffiffi
g11
p ¼ ffiffiffiffiffiffig33p ¼ 1, ffiffiffiffiffiffig22p ¼ r, then
∇ϕ ¼ 1ffiffiffiffiffi
gii
p ∂ϕ
∂xi
gk
it follows that
∇ϕ ¼ ∂ϕ
∂r
gr þ
1
r
∂ϕ
∂θ
gθ þ
∂ϕ
∂z
gz
Exercise 4.4 Calculate the gradient of the scalar function ϕ(x1) expressed in
spherical coordinates.
For the spherical coordinates
ffiffiffiffiffiffi
g11
p ¼ 1, ffiffiffiffiffiffig22p ¼ r, ffiffiffiffiffiffig33p ¼ r sinϕ, then
∇ϕ ¼ ∂ϕ
∂r
gr þ
1
r
∂ϕ
∂ϕ
gϕ þ
1
r sin ϕ
∂ϕ
∂xθ
gθ
Exercise 4.5 Show that
∂2ϕ xið Þ
∂xi∂xj
is a second-order tensor, where ϕ(xi) is a scalar
function.
Putting
Tij ¼ ∂
2ϕ
∂xi∂xj
¼ ϕ, ij
for the coordinate system X
i
∂2ϕ
∂xi∂xj
¼ ∂ϕ
∂xi
∂ϕ
∂xk
∂xk
∂xk
� �
¼ ∂ϕ
∂xm
∂ϕ
∂xk
∂xk
∂xj
� �
 �
∂xm
∂xi
¼ ∂x
k
∂xj
∂xm
∂xi
� �
∂2ϕ
∂xm∂xk
This transformation law proves that
∂2ϕ xið Þ
∂xi∂xj
is a second-order tensor.
Exercise 4.6 Calculate the directional derivative of the function
ϕ x, yð Þ ¼ x2 þ y2 � 3xy3, at the point P(1; 2) in the direction of vector
u ¼ 1
2
e1 þ
ffiffi
3
p
2
e2, being e1(1; 0), e2(0; 1).
The gradient of the scalar field is given by
∇ϕ ¼ 2x� 3y3� �e1 þ 2y� 9xy2� �e2
4.2 Gradient 173
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and in point P(1; 2)
∇ϕ ¼ �22e1 � 32e2
The vector u is a unit vector, for uuk k ¼ 12 e1 þ
ffiffi
3
p
2
e2, whereby it follows that for the
directional derivative
∇ϕ � u ¼ �22� 1
2
� 32�
ffiffiffi
3
p
2
¼ �11� 16
ffiffiffi
3
p
4.3 Divergence
The analysis of the flow magnitude for the vectorial field u that passes through the
volume V involved by the closed surface S with respect to this volume leads to the
conception of a differential operator (Fig. 4.5a). In volume V let the elementary
parallelepiped with sides dx1, dx2, dx3, and the vectorial function u continuous with
continuous partial derivatives (Fig. 4.5b).
The study of the flow of u that passes through the volume V with respect to this
volume is carried out considering the point P(x1; x2; x3) at the center of an elemen-
tary parallelepiped (Fig. 4.5). For the face dS1, with upward normal unit vector n
(1; 0; 0), the component of u in the direction of axis OX1 is given by
u � n ¼ u1
1X
O
n
n
n
2X
12S
*
12S
P
1dx
2dx
3dx
u u
3X
V
S
P
a b
Fig. 4.5 Flow of the vectorial function u: (a) that passes through the volume V and (b) in the
elementary parallelepiped
174 4 Differential Operators
The center of the elementary area dS1 ¼ dx1dx2 has coordinates x1 þ dx12 ;
�
dx2; dx3Þ, whereby it follows that for the surface integral in this face of the
elementary parallelepiped dVðð
dS1
u � ndS ffi u1 x1 þ dx
1
2
; dx2; dx3
� �
dx2dx3
for the area considered is elementary, which allows calculating approximately the
surface integral as the dot product
u:ndS ¼ u1dx2dx3
For the elementary face dS*1 ¼ dx2dx3 with upward normal unit vector
n �1; 0; 0ð Þ centered in the midpoint x1 � dx1
2
; dx2; dx3
� �
it follows that in an
analogous way ðð
dS*1
u:ndS ffi �u1 x1 � dx
1
2
; dx2; dx3
� �
dx2dx3
Adding these contributionsðð
dS1þdS*1
u:ndS ffi u1 x1 þ dx
1
2
; dx2; dx3
� �
� u1 x1 � dx
1
2
; dx2; dx3
� �
 �
dx2dx3
ffi u1 x1� �dx2dx3
The component u1 in the point of coordinate dx1 varies according to the rate
du1 ¼ ∂u
1
∂x1
dx1
then ðð
dS1þdS*1
u � ndS ffi ∂u
1
∂x1
dx1dx2dx3 ffi ∂u
1
∂x1
dV
and the same way for the components u2 and u3 in the other faces of the parallel-
epiped the result is, respectively
4.3 Divergence 175
ðð
dS2þdS*2
u � n dS ffi ∂u
2
∂x2
dV
ðð
dS3þdS*3
u � ndS ffi ∂u
3
∂x3
dV
Adding these three expressions results for the six faces of the elementary
parallelepiped ðð
S
u � n dS ¼
ððð
V
∂u1
∂x1
þ ∂u
2
∂x2
þ ∂u
3
∂x3
� �
dV
Putting
divu ¼ ∂u
1
∂x1
þ ∂u
2
∂x2
þ ∂u
3
∂x3
ð4:3:1Þ
this analysis leads to the following definition for the divergence of the vectorial
function u at point P(xi)
divu ¼ lim
V!0
1
V
ðð
S
u � ndS ð4:3:2Þ
that can be interpreted as the dot product between the nabla operator and the
vectorial function u, thus
divu ¼ ∇ � u ¼ ∂� � �
∂xj
� �
gj � uigi
To demonstrate that expression (4.3.2) represents the divergence of the vectorial
function u, consider the sphere of radius R > 0, of surface S(R) and volume V(R),
centered at point P located in the vectorial space E3. For the vectorial field u acting
in the space
divu Pð Þ ¼ lim
R!0
1
V Rð Þ
ðð
S Rð Þ
u � ndS ð4:3:3Þ
Let g Pð Þ ¼ divu, and admitting that g(xi) is a continuous function that can be
written as
g xi
� � ¼ g Pð Þ þ h xi� �
where
h xi
� �
xi!P ¼ 0
176 4 Differential Operators
Applying the divergence theorem to the vectorial field
1
V Rð Þ
ðð
S Rð Þ
u:n dS ¼ 1
V Rð Þ
ððð
V Rð Þ
h xi
� �
dV ¼ 1
V Rð Þ
ððð
V Rð Þ
h Pð ÞdV þ 1
V Rð Þ
ððð
V Rð Þ
h xi
� �
dV
As g Pð Þ ¼ divu:
1
V Rð Þ
ððð
V Rð Þ
g Pð ÞdV ¼ 1
V Rð Þ g Pð Þ
ððð
V Rð Þ
dV ¼ 1
V Rð Þ g Pð ÞV Rð Þ ¼ g Pð Þ
For the function h(xi) the result when R ! 0 is
1
V Rð Þ
ððð
V Rð Þ
h xi
� �
dV
�������
������� ¼ Maxxi�Pk k	R h xi
� ��� �� 1
V Rð Þ
ððð
V Rð Þ
dV 	 Max
xi�Pk k	R
h xi
� ��� ��
The maximum value of this function fulfills the condition h xið Þk k ! 0 when
xi � Pk k ! 0, then the expression (4.3.2) represents the divergence of the vectorial
function u at the point P.
This expression is valid for any kind of coordinate system, which shows that the
divergence is independent of the referential system. This analysis was formulated
for a Cartesian coordinate system for a question of simplicity, being that for the case
of curvilinear coordinate systems it was enough to adopt the local trihedron with
unit vectors (g1; g2; g3), and one elementary parallelepiped of volume dV.
The scalar field generated by the applying of the divergence to the vectorial field
defined by the vectorial function u is called solenoidal or vorticular field, when
divu ¼ 0, where u is a solenoidal vector, and this field is called field without
source.
4.3.1 Divergence Theorem
The divergence allows writing the Gauß-Ostrogradsky theorem asðð
S
u � ndS ¼
ððð
V
∇ � udV ð4:3:4Þ
which is called the divergence theorem. The symbology adopted in expression
(4.3.4) does not change the characteristics and properties shown in item 3.4.
Let a solenoidal field acting in a region R be located between the two closed
surfaces S1 and S2 (Fig. 4.6), then
4.3 Divergence 177
ðð
S1
u � ndS ¼
ðð
S2
u � ndS
To demonstrate this equality consider the closed surface S1 with upward normal
unit vector n, with R to the left of the outline of this surface that involves the volume
V1, and the closed surface S2 with unit downward unit normal vector n, involving
the volume V2. Applying the divergence theorem
divu ¼
ðð
S1
u � ndS�
ðð
S2
u � ndS ¼ 0 )
ðð
S1
u � n dS ¼
ðð
S2
u � ndS
then it is enough to calculate only the integral of a surface.
For a field represented by the vectorial function u ¼ ϕ∇ψ , where ϕ and ψ are
scalar functions
∇ � u ¼ ∇ ϕ∇ψð Þ ¼ ϕ∇ �∇ψ þ∇ϕ �∇ψ ¼ ϕ∇2ψ þ∇ϕ �∇ψ
The component of u in the direction of the normal unit vector n is given by
u � n ¼ ϕn �∇ψ ¼ ϕ∂ψ
∂n
and applying the divergence theoremðð
S
u � ndS ¼
ððð
V
∇ � udV
1S
2S
21 VVR −=
1V2V
Fig. 4.6 Solenoidal field in
a region R between two
volumes
178 4 Differential Operators
results in ðð
S
ϕ
∂ψ
∂n
dS ¼
ððð
V
ϕ∇2ψ þ∇ϕ �∇ψ� �dV
that is called Green’s first formula.
If the vectorial function is given by u ¼ ϕ∇ψ þ ψ∇ϕ, in an analogous way
∇ � u ¼ ϕ∇2ψ þ ψ∇2ϕ
u � n ¼ ϕ∂ψ
∂n
� ψ ∂ϕ
∂n
whereby ðð
S
ϕ
∂ψ
∂n
� ψ ∂ϕ
∂n
dS ¼
ððð
V
ϕ∇2ψ þ ψ∇2ϕ� �dV
that is called Green’s second formula.
4.3.2 Contravariant and Covariant Components
The vectorial function u can be expressed by means of their contravariant or
covariant components, so it is necessary to calculate this function’s divergence
for these components. For the vector’s contravariant coordinates
divu ¼ ∇ � u ¼ g j � ∂u
igi
∂xj
� �
ð4:3:5Þ
and for its covariant coordinates
divu ¼ ∇ � u ¼ gj �
∂uigi
∂xj
� �
ð4:3:6Þ
The terms in parenthesis in theseexpressions indicate that this definition can be
amplified considering the vector’s covariant derivatives, expressed in their
contravariant and covariant coordinates.
Let the covariant derivative of the contravariant vector ui be:
∂ku
i ¼ ∂u
i
∂xk
þ u jΓ ijk
4.3 Divergence 179
that generates a tensor which contraction for i ¼ k provides
∂iu
i ¼ ∂u
i
∂xi
þ ujΓ iji
and rewriting the expression (2.4.23)
Γ iji ¼
∂ ‘n
ffiffiffi
g
p� �
∂xj
¼ 1ffiffiffi
g
p ∂
ffiffiffi
g
p� �
∂xj
The use of this expression is more adequate, for it abbreviates the calculation of
the Christoffel symbol. Substituting this expression in the previous expression
∂iu
i ¼ ∂u
i
∂xi
þ u j 1ffiffiffi
g
p ∂
ffiffiffi
g
p� �
∂xj
¼ 1ffiffiffi
g
p ffiffiffigp ∂ui
∂xi
þ u
jffiffiffi
g
p ∂
1
2
‘ng
� �
∂xj
 �
and replacing the indexes i ! j of the first term to the right
∂iu
i ¼ 1ffiffiffi
g
p ffiffiffigp ∂u j
∂xj
þ u
j
2
ffiffiffi
g
p ∂
1
2
‘ng
� �
∂xj
 �
or in a compact form
∂iu
i ¼ 1ffiffiffi
g
p ∂
ffiffiffi
g
p
uj
� �
∂xj
ð4:3:7Þ
It is verified that expression (4.3.7), deducted by means of the contravariant
vector ui, represents a scalar, for it was obtained by means of contraction of the
second-order tensor. The other way of formulating this analysis is by means of the
covariant derivative of their covariant components.
Let the covariant derivative of the covariant vector ui be:
∂iui ¼ ∂i gijuj
� �
that developed leads to the following expression
∂iui ¼ ∂i gij
� �
uj þ gij∂i uj
� �
Ricci’s lemma shows that ∂i gijð Þ ¼ 0 whereby
∂iui ¼ gij∂i uj
� � ¼ ∂iu j
180 4 Differential Operators
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and the contraction of this tensor for i ¼ j provides
∂iui ¼ ∂iui ¼ divui ¼ 1ffiffiffi
g
p ∂
ffiffiffi
g
p
ui
� �
∂xi
ð4:3:8Þ
Expressions (4.3.7) and (4.3.8) provide the same result, i.e., ∂iui ¼ ∂iui. Then
the covariant derivative of a vector is independent of the type of the component.
The divergence defined by expressions (4.3.5) and (4.3.6) is the dot product of
the nabla operator by the vector to which it is applied. The development of the
derivatives indicated in these expressions leads to the same results of expressions
(4.3.7) and (4.3.8), whereby these last expressions represent the divergence of a
vectorial function.
For the Cartesian coordinates
∇ � u ¼ ∂ui
∂xi
ð4:3:9Þ
4.3.3 Orthogonal Coordinate Systems
Consider the elementary parallelepiped with sides ds1, ds2, ds3, defined in the
curvilinear orthogonal coordinates OXj, by means of which the flow of the field is
represented by the vectorial function u (Fig. 4.7).
The divergence of this field is given by
∇ � u ¼ lim
V!0
1
V
ðð
S
u � ndS
Let
dsi ¼ hidxi
O
1X
2X
3X
1ds 2ds
3ds
( )321 ;; xxxu
2gn =
( )3221 ;; xxxx ∂+u2gn −=
Fig. 4.7 Divergence of the vectorial function u in the curvilinear orthogonal coordinates
4.3 Divergence 181
dV ¼ ds1ds2ds3 ¼ h1h2h3dx1dx2dx3
there is, respectively, for the face with upward normal unit vector n ¼ �g2 and
n ¼ g2
�u � g2ds1ds3 ¼ �u2h1h3dx1dx3 ¼ �u2 h1h3 þ
∂ u2h1h3ð Þ
∂x2
dx2
 �
dx1dx3
In an analogous way, for the other faces
�u � g1ds2ds3 ¼ �u1h2h3dx2dx3 ¼ �u1 h2h3 þ
∂ u1h2h3ð Þ
∂x1
dx2
 �
dx2dx3
�u � g2ds1ds3 ¼ �u2h1h3dx1dx3 ¼ �u2 h1h3 þ
∂ u2h1h3ð Þ
∂x2
dx1
 �
dx1dx3
�u � g3ds1ds2 ¼ �u3h1h2dx1dx2 ¼ �u3 h1h2 þ
∂ u3h1h2ð Þ
∂x3
dx3
 �
dx1dx2
Adding the expressions relative to the six faces of the parallelepipedðð
S
u � ndS ¼ ∂ u
1h2h3ð Þ
∂x1
þ ∂ u
2h1h3ð Þ
∂x2
þ ∂ u
3h1h2ð Þ
∂x3
 �
dx1dx2dx3
but
dx1dx2dx ¼ dV
h1h2h3
then
∇ � u ¼ lim
V!0
ðð
S
u � ndS ¼ 1
h1h2h3
∂ u1h2h3ð Þ
∂x1
þ ∂ u
2h1h3ð Þ
∂x2
þ ∂ u
3h1h2ð Þ
∂x3
 �
The result for the orthogonal coordinate system is
∇ � u ¼ divu ¼ 1
h1h2h3
∂
∂xi
h1h2h3u
i
hi
� �
ð4:3:10Þ
where hi ¼ ffiffiffiffiffiffiffiffig iið Þp are the components of the metric tensor, and the indexes in
parenthesis do not indicate summation.
182 4 Differential Operators
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4.3.4 Physical Components
With expression (4.3.6) the physical components of the divergence of a vector takes
the form
∇ � u* ¼ divu* ¼ 1
h1h2h3
∂
∂xi
h1h2h3u
*i
hi
� �
ð4:3:11Þ
where u* i are the vector’s physical components.
4.3.5 Properties
As the divergence is the dot product of the nabla operator for a vectorial function,
the distributive property of the dot product is valid. For the sum of two vectorial
functions u and v:
div uþ vð Þ ¼ ∇ � uþ vð Þ ¼ ∇ � uþ∇ � v ð4:3:12Þ
and in terms of the covariant derivative
∇ � uþ vð Þ ¼ ∂iui þ ∂ivi ð4:3:13Þ
and for the Cartesian coordinates the result is
∇ � uþ vð Þ ¼ ∂u
i
∂xi
þ ∂v
i
∂xi
ð4:3:14Þ
Considering the vectorial function mu, where m is a scalar, the result of the dot
product of vectors is
div muð Þ ¼ ∇ � muð Þ ¼ m∇ � u ð4:3:15Þ
These two demonstrations prove that the divergence, for these cases, is a linear
operator. In general, the divergence is not a linear operator, as it will be shown in
Exercise 4.7.
4.3.6 Divergence of a Second-Order Tensor
The generalization of the divergence theorem for tensorial fields is immediate.
Consider, for example, the field represented by the tensorial function of the second
4.3 Divergence 183
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order T(r) in space E3, which components depend on the position vector, i.e.,
Tij ¼ Tij rð Þ. For the surface S smooth and continuous by parts in its two faces,
with normal unit vector n(n1; n2; n3) varying on each point of the surface, the flow
of this tensorial function through S is given by the vector v of components
vi ¼
ðð
S
TijnjdS; i, j ¼ 1, 2, 3
or
vi ¼
ðð
S
TjinjdS; i, j ¼ 1, 2, 3
In absolute notation for the flow v the result is
v ¼
ðð
S
T� ndS ð4:3:16Þ
The flow of the unit tensor δij through the closed surface S is given by the
components of vector n:
vi ¼
ðð
S
δijnjdS ¼
ðð
S
nidS
or in absolute notation
vi ¼
ðð
S
ndS
The comparison with expression (4.2.3)
∇ϕ xi
� � ¼ lim
V!0
1
V
ðð
S
ϕ xi
� �
ndS
shows that ϕ xið Þ ¼ 1, i.e., ϕ xið Þ ¼ constant, so ∇ϕ xið Þ ¼ 0, which indicates that
v ¼ 0. Concluding that for a unitary tensorial field the flow through the closed
surface S is null.
The concept of a field’s divergence can be extended to the tensorial fields, for it
is enough that the tensors be contravariant. In the case of covariant tensors their
indexes must be raised by means of the metric tensor, next they must be derived and
contracted.
184 4 Differential Operators
There are distinct divergences, depending on the index to be contracted. For Tij
there are two divergences:∂iTij and∂jTij. If the tensor is symmetrical Tij ¼ Tji then
∂iTij ¼ ∂jTij, i.e., the divergence is unique.
The divergence components of a contravariant second-order tensor are given by
divTij ¼ ∂jTij ð4:3:17Þ
and the covariant derivative of the components for this tensor is
∂kT
ij ¼ ∂T
ij
∂xk
þ TmkΓ imk þ TimΓ jmk
� �
With k ¼ j:
∂jT
ij ¼ ∂T
ij
∂xj
þ TmjΓ imj þ TimΓ jmj
� �
being
Γ jmj ¼
∂ ‘n
ffiffiffi
g
p� �
∂xm
it follows that
∂jT
ij ¼ ∂T
ij
∂xj
þ TmjΓ imj þ Tim
∂ ‘n
ffiffiffi
g
p� �
∂xm
 �
The change of the indexes m ! j in the last term in brackets provides
∂kT
ij ¼ ∂T
ij
∂xj
þ TmjΓ imj þ Tij
∂ ‘n
ffiffiffi
g
p� �
∂xj
 �
or
∂kT
ij ¼ TmjΓ imj þ
1ffiffiffi
g
p ffiffiffigp ∂Tij
∂xj
þ Tij ∂
ffiffiffi
g
p� �
∂xj
 �
then
divTij ¼ TmjΓ imj þ
1ffiffiffi
g
p ∂
ffiffiffi
g
p
Tij
� �
∂xk
ð4:3:18Þ
that shows that the divergence of a second-order tensor is a vector.
4.3 Divergence 185
For a mixed second-order tensor the result is
divT ij ¼ ∂iT ij
and rewriting expression (2.5.21)
∂kT
i
j ¼
∂T ij
∂xk
þ Tmj Γ imk � T imΓmjk
Assuming i ¼ k:
∂iT
i
j ¼
∂T ij
∂xk
þ Tmj Γ imi � T imΓmji
and with
Γ imi ¼
∂ ‘n
ffiffiffi
g
p� �
∂xm
then
∂iT
i
j ¼
∂T ij
∂xk
þ Tmj
∂ ‘n
ffiffiffi
g
p� �
∂xm
� T imΓmji
" #
The change of indexes m ! i in the second term in brackets provides
∂iT
i
j ¼
∂T ij
∂xk
þ T ij
∂ ‘n
ffiffiffi
g
p� �
∂xi
� T imΓmji
" #
or∂iT
i
j ¼
ffiffiffi
g
p ∂T ij
∂xk
þ 1ffiffiffi
g
p T ij
∂
ffiffiffi
g
p� �
∂xi
� T imΓmji
then
∂iT
i
j ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� T imΓmji ð4:3:19Þ
The generalization of the definition of the divergence for a third-order tensor is
immediate
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∇ � T ¼ g j � ∂jT ¼ g j � ∂jTkimgk � gi � gm ¼ ∂jTkim
� �
g j � gk � gi � gm
¼ ∂jTkim
� �
δ jkgi � gm
thus
∇ � T ¼ ∂jTjimgi � gm ð4:3:20Þ
This expression shows that ∇ � T is a second-order tensor.
For a tensor T of order p then ∇ � T is a tensor of order p� 1ð Þ. In absolute or
invariant notation the result for the divergence of tensor T is
∇ � T ¼ ∇� Tð Þ�G ð4:3:21Þ
where G is the metric tensor.
In the particular case in which divT ¼ 0 the tensor T defines a tensorial
solenoidal field.
The divergence theorem also applies to a tensorial field. Let the field be defined
by u ¼ Tv, which in terms of the components of vectors and tensor is given by
ui ¼ Tikvk, being v an arbitrary and constant vector. Applying the divergence
theorem to this field ðð
S
u � ndS ¼
ððð
V
∇ � udV
where
∇ � u ¼ ∇ Tvð Þ ¼ ∇T � v
This vector has components
∂Tik
∂xk
vi, and the component of vector u in the
direction of the normal unit vector n is given by the dot product
u � n ¼ Tikvið Þnk
then ðð
S
Tikvið Þnk dS ¼
ððð
V
∂Tik
∂xk
vidV
whereby in terms of the tensor components the result isðð
S
Tik nk dS ¼
ððð
V
∂Tik
∂xk
dV ð4:3:22Þ
4.3 Divergence 187
and in absolute notation this expression becomesðð
S
T� ndS ¼
ððð
V
∇ � TdV ð4:3:23Þ
Exercise 4.7 Calculate: (a) ∇ � ϕuð Þ; (b) ∇ � u� vð Þ.
(a) The field divergence defined by the product of a scalar function ϕ(xi) by a
vector u is given by
div ϕuð Þ ¼ ∇ � ϕuð Þ ¼ gm ∂ ϕu
kgk
� �
∂xm
¼ gm ∂ϕ
∂xm
ukgk þ ϕ
∂uk
∂xm
gk þ ϕuk
∂gk
∂xm
� �
and substituting (2.3.10)
∂gk
∂xm
¼ Γ pkmgp
in the previous expression
div ϕuð Þ ¼ gm ∂ϕ
∂xm
ukgk þ ϕ
∂uk
∂xm
gk þ ϕukΓ pkmgp
� �
The permutation of the indexes p $ k in the third member in parenthesis
provides
div ϕuð Þ ¼ ∂ϕ
∂xm
uk þ ϕ ∂u
k
∂xm
þ ϕupΓ kpm
� �
gm � gk ¼
∂ϕ
∂xm
uk þ ϕ ∂u
k
∂xm
þ ϕupΓ kpm
and with
∂mu
k ¼ ∂ϕ
∂xm
uk þ ϕ ∂u
k
∂xm
) div ϕuð Þ ¼ ∂ϕ
∂xm
uk þ ϕ∂muk
Putting
∂mu
k ¼ ∇uk ∂ϕ
∂xm
¼ ∇ϕ
thus
∇ � ϕuð Þ ¼ ∇ϕð Þ � uþ ϕ ∇uð Þ
188 4 Differential Operators
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or
div ϕuð Þ ¼ gradϕ � uþ ϕgradu
In this case the divergence is not a linear operator, but for ϕ ¼ m, where m is a
constant, the result is expression (4.3.15), verifying the linearity of this
operator.
(b) The field represented by the vectorial function generated by the cross product
w ¼ u� v is given by
w ¼ wpgp ¼ εpqruqvrgp
The divergence of this function is given by
∇ � u� vð Þ ¼ gi � ∂ w
pgp
� �
∂xi
¼ ∂w
p
∂xi
gp þ wp
∂gp
∂xi
� �
� gi
and the expression
∂gp
∂xi
¼ Γ jpigj
substituted in the previous expression provides
∇ � u� vð Þ ¼ ∂w
p
∂xi
gp þ wpΓ jpigj
� �
� gi
Interchanging indexes p $ j in the second term in parenthesis it follows that
∇ � u� vð Þ ¼ ∂w
p
∂xi
þ wjΓ pji
� �
gi � gp ¼ δ ip∂iwp
or
∂pw
p ¼ ∂p εpqruqvr
� � ¼ ∂pεpqr� �uqvr þ εpqr∂p uqvr� �
and with
∂pw
p ¼ εpqr∂p uqvr
� �
thus
∇ � u� vð Þ ¼ εpqr∂p uqvr
� �
4.3 Divergence 189
whereby
∇ � u� vð Þ ¼ εpqr ∂puq
� �
vr þ uq∂pvr
� �
With the εpqr ¼ εqpr and εpqr ¼ �εrpq the results for the terms to the right are
εpqr ∂puq
� �
vr ¼ εrpq ∂puq
� �
vr ¼ v �∇u
εpqruq ∂pvr
� � ¼ �εqpruq ∂pvr� � ¼ �u �∇v
whereby
∇ � u� vð Þ ¼ v �∇� u� u �∇� v ) div u� vð Þ ¼ v � rotu� u � rotv
For the Cartesian coordinates
∇ � u� vð Þ ¼ εijk
∂ ujvk
� �
∂xi
Exercise 4.8 Let T ij and Tij be associated tensors, write div T
i
j in terms of the
symmetrical tensor Tij.
The divergence of a second-order tensor is given by
divT ij ¼ ∂iT ij ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� T imΓmji
and with
Γmij ¼ gmkΓij,k
thus
divT ij ¼ ∂iT ij ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� T imgmkΓij,k
Let
Γij,k ¼ 1
2
∂gjk
∂xi
þ ∂gik
∂xj
þ ∂gij
∂xk
� �
T img
mk ¼ Tik
then
∂iT
i
j ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� Tik1
2
∂gjk
∂xi
þ ∂gik
∂xj
þ ∂gij
∂xk
� �
190 4 Differential Operators
or
∂iT
i
j ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� 1
2
Tik
∂gjk
∂xi
� 1
2
Tik
∂gik
∂xj
� 1
2
Tik
∂gij
∂xk
Interchanging the indexes i $ j in the last term to the right
∂iT
i
j ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� 1
2
Tik
∂gjk
∂xi
� 1
2
Tik
∂gik
∂xj
� 1
2
Tki
∂gkj
∂xk
As
gjk ¼ gkj Tik ¼ Tki
thus
∂iT
i
j ¼
1ffiffiffi
g
p
∂ T ij
ffiffiffi
g
p� �
∂xi
� 1
2
Tik
∂gik
∂xj
Exercise 4.9 Calculate the divergence of vector ui expressed in cylindrical
coordinates.
For the cylindrical coordinates
ffiffiffi
g
p ¼ r, and with the contravariant components
of vector (ur, uθ, uz) it follows that
divui ¼ 1ffiffiffi
g
p ∂
ffiffiffi
g
p
ui
� �
∂xi
¼ 1
r
∂ ruið Þ
∂xi
 �
¼ 1
r
∂ rurð Þ
∂r
þ ∂ ru
θ
� �
∂θ
þ ∂ ru
zð Þ
∂z
 �
¼ ∂u
r
∂r
þ ∂u
θ
∂θ
þ ∂u
z
∂z
þ ur
In an analogous way in terms of the vector’s covariant components
divuj ¼ ∂ur∂r þ
1
r2
∂uθ
∂θ
þ ∂uz
∂z
þ ur
Exercise 4.10 Calculate the divergence of vector ui expressed in spherical
coordinates.
For the spherical coordinates
ffiffiffi
g
p ¼ r2 sinϕ, and with the contravariant compo-
nents of vector (ur, uϕ, uθ) it follows that
4.3 Divergence 191
divui ¼ 1ffiffiffi
g
p ∂
ffiffiffi
g
p
ui
� �
∂xi
¼ 1
r2 sinϕ
∂ r2 sin ϕuið Þ
∂xi
 �
¼ 1
r2 sinϕ
∂ r2 sinϕurð Þ
∂r
þ ∂ r
2 sinϕuϕ
� �
∂ϕ
þ ∂ r
2 sin ϕuθ
� �
∂θ
 �
¼ ∂u
r
∂r
þ ∂u
ϕ
∂ϕ
þ ∂u
θ
∂θ
þ 2u
r
r
þ cotϕð Þuϕ
For the vector’s covariant components the result is
divuj ¼ ∂ur∂r þ
1
r2
∂uϕ
∂ϕ
þ 1
r2 sin 2ϕ
∂uθ
∂θ
þ 2ur
r
þ cotϕð Þ
r2
uϕ
Exercise 4.11 Let r be the position vector of the points in the space E3, show that:
(a) div r ¼ 3; (b) div rn rð Þ ¼ nþ 3ð Þ rn; (c) div rr3
� � ¼ 0; (d) div rr� � ¼ 2r.
(a) With the definition of divergence
∇ � r ¼ gi
∂
∂xi
� �
� r ¼ gi �
∂r
∂xi
but
∂r
∂xi
¼ gi
then
∇ � r ¼ gi � gi
For i ¼ 1, 2, 3 the result is
div r ¼ 3 Q:E:D:
(b) Let
div ϕuð Þ ¼ ϕdivuþ ugradϕ
and putting
u ¼ r ϕ ¼ rn
thus
div rnrð Þ ¼ rn div rþ rgrad rn ¼ 3rn þ r � nrn�11
r
r
� �
¼ 3rn þ nrn�2r2� �
192 4 Differential Operators
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then
div rn rð Þ ¼ nþ 3ð Þ rn Q:E:D:
(c) Putting
div
r
r3
� �
¼ div r�3r� �
it follows that
div r�3rð Þ ¼ r�3div rþ r � grad r�3 ¼ 3r�3 þ r � �3r�4grad rð Þ ¼ 3r�3 þ r � �3r�4r
r
� �
¼ 3r�3 þ r2 �3r�41
r
� �
whereby
div r�3r
� � ¼ 0 Q:E:D:
This conclusion shows that r�3r is a solenoidal vectorial function.
(d) Putting
div
r
r
� �
¼ div 1
r
r
� �
r ¼ xiþ yjþ zk
it follows that
div
1
r
r
� �
¼ div x
r
iþ y
r
jþ z
r
k
� �
¼ ∂
∂x
x
r
� �
þ ∂
∂y
y
r
� �
þ ∂
∂z
z
r
� �
¼ 1
r
� x
r2
∂r
∂x
� �
þ 1
r
� y
r2
∂r
∂y
� �
þ 1
r
� z
r2
∂r
∂z
� �
and with
r2 ¼ x2 þ y2 þ z2
∂r
∂x
¼ x
r
∂r
∂y
¼ y
r
∂r
∂z
¼ z
r
thus
div
1
r
r
� �
¼ 3
r
� x
r2
x
r
þ y
r2
y
r
þ z
r2
z
r
� �
4.3 Divergence 193
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whereby
div
r
r
� �
¼ 2
r
Q:E:D:
4.4 Curl
The vector product of the nabla operator by a vector generates a differential
operator linked to the direction of rotation of the coordinate system defining the
curl, also called rotation or whirl. In absolute notation this operator is written as
∇� u ¼ rot u ¼ v ð4:4:1Þ
In English literature the notation curl u is used to designate rotational of vector u,
which was adopted firstly by Maxwell. The term curl literally means ring, and it
designates the pseudovector ∇� u. With
v ¼ gi � ∂ ujg
j
� �
∂xi
¼ gi � ∂uj
∂xi
g j þ gi � uj ∂g
j
∂xi
and rewriting expression (2.4.4)
∂g j
∂xi
¼ �Γ jkigm
it follows for the second term of the member to the right of expression (4.4.1)
ujg
i � ∂g
m
∂xi
¼ �ujΓ jkigi � gm
The cross product of these vectors is given by
gi � gm ¼ e
imkffiffiffi
g
p gk ¼
þ1 for an even number of
permutationsof the indexes
�1 for an odd number of
permutations of the indexes
0 when there are repeated indexes
8>>>>>><>>>>>>:
and substituting results in
ujg
i � ∂g
m
∂xi
¼ � ujffiffiffi
g
p Γ jki � Γ jik
� �
gk ¼ 0
194 4 Differential Operators
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whereby
∇� u ¼ v ¼ ∂uj
∂xi
gi � gm
As
gi � g j ¼ e
ijkffiffiffi
g
p gk
in tensorial terms
∇� u ¼ e
ijkffiffiffi
g
p ∂uj
∂xi
gk ð4:4:2Þ
As a function of Ricci’s pseudotensor
∇� u ¼ εijk ∂uj
∂xi
gk ð4:4:3Þ
and with
εijk ¼ e
ijkffiffiffi
g
p
the expression (4.4.3) after a cyclic permutation of the indexes i, j, k ¼ 1, 2, 3 takes
the form
∇� u ¼ 1ffiffiffi
g
p ∂uj
∂xi
� ∂ui
∂xj
� �
gk ð4:4:4Þ
In a space provided with metric, the curl of a vectorial function can also be
defined by means of its contravariant components, for these relate with its covariant
components by means of the metric tensor.
In an analogous way, the results for the contravariant coordinates are
∇� u ¼ g‘ � ∂ u
kgk
� �
∂x‘
¼ g‘ � ∂u
k
∂x‘
gk þ uk
∂gk
∂x‘
� �
∂gk
∂x‘
¼ Γmk‘gm
∇� u ¼ g‘ � ∂u
k
∂x‘
gk þ ukΓmk‘gm
� �
¼ ∂u
k
∂x‘
þ umΓ km‘
� �
g‘ � gk
4.4 Curl 195
∇� u ¼ ∂‘uk g‘ � gk
g‘ ¼ g‘jgj
∇� u ¼ ∂‘uk g‘jgj � gk
gj � gk ¼ εijkgi
∇� u ¼ εijk ∂‘uk
� �
g‘jgi ð4:4:5Þ
A vectorial field is called an irrotational field when ∇� u ¼ 0, then
∂uj
∂xi
¼ ∂ui
∂xj
In space E3 the curl∇� u is an axial vector (vectorial density), so it is associated
to an antisymmetric second-order tensor, which components are
Aij ¼
0
∂u2
∂x1
� ∂u
1
∂x2
∂u3
∂x1
� ∂u
1
∂x3
∂u1
∂x2
� ∂u
2
∂x1
0
∂u3
∂x2
� ∂u
2
∂x3
∂u1
∂x3
� ∂u
3
∂x1
∂u2
∂x3
� ∂u
3
∂x2
0
26666664
37777775 ð4:4:6Þ
For the space EN the curl ∇� u has 12N N � 1ð Þ independent components. In
space E2 the curl is a pseudoscalar. For the Cartesian coordinates
∇� u ¼ eijk ∂uj∂xi gk ð4:4:7Þ
or in a determinant form
∇� u ¼
i j k
∂
∂x1
∂
∂x2
∂
∂x3
u1 u2 u3
							
							 ð4:4:8Þ
4.4.1 Stokes Theorem
In expression (3.3.7) with F ¼ u, and with the expression (4.4.8) Stokes theorem in
vectorial notation is given byðð
S
n �∇� u dS ¼
þ
C
u � dr ð4:4:9Þ
196 4 Differential Operators
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A more consistent definition of the curl can be formulated analyzing the circu-
lation of the vectorial field u in a closed surface S with upward unit normal vector n
(Fig. 4.8a).
Consider the elementary rectangle dS determined in the orthogonal curvilinear
coordinate system X
j
, with sides h1dx
1 and h2dx
2 located in the plane OX
2
X
3
, with
the point P x1; x2; x3
� �
located in its center. Locally, the coordinate system X
j
is
considered as a Cartesian orthogonal system (Fig. 4.8b), with the scale factors
hi, i ¼ 1, 2, 3.
The line integral
þ
C
u � dr along the perimeter of this rectangle is carried out
dividing this perimeter into segments C1,C2,C3,C4. The center of segment C1 of
perimeter of the rectangle is given by the coordinates x1; x2; x3 � dx3
2
� �
then
u � dr ¼ u2dx2. As this length is elementary its contribution to the line integral is
given by þ
C1
u � dr ffi u2 x1; x2; x3 � dx
3
2
� �
h2dx
2
For segment C3 with center x
1; x2; x3 þ dx3
2
� �
:
þ
C3
u � dr ffi �u2 x1; x2; x3 þ dx
3
2
� �
h2dx
2
where the negative sign indicates that the direction of the path is contrary to the
coordinate axis.
n
O
P
3C
4C
iS
P i
C
S
3X
1X
2X
1g
2g
3g
3
32 xdhC =
2
21 xdhC =
a b
Fig. 4.8 Concept of curl: (a) circulation in a closed surface and (b) elementary rectangle
4.4 Curl 197
Adding the contributions of these two segmentsþ
C1þC3
u � dr ffi u2 x1; x2; x3 � dx
3
2
� �
� u2 x1; x2; x3 þ dx
3
2
� �
 �
h2dx
2
The component u2 varies according to the rate
du2 ¼ �∂u
2
∂x3
dx3
where the negative sign indicates that this variation decreases in the positive
direction of axis OX
1
, it follows thatþ
C1þC3
u � dr ffi �∂u
2
∂x3
dx3h2dx
2
and dividing by dS ¼ h2h3dx2dx3
1
dS
þ
C1þC3
u � dr ffi � 1
h2h3
∂ u2h2ð Þ
∂x3
Adopting analogous formulations for the other segmentsþ
C2þC4
u � dr ffi u3 x1; x2; x3 þ dx
3
2
� �
� u3 x1; x2; x3 � dx
3
2
� �
 �
h3dx
3 ffi ∂u
3
∂x2
dx2h3dx
3
1
dS
þ
C2þC4
u � dr ffi 1
h2h3
∂ u3h3ð Þ
∂x2
Adding these contributions the result when dS ! 0 is
∂ h3u3ð Þ
∂x2
� ∂ h2u
2ð Þ
∂x3
 �
¼ lim
dS!0
1
dS
þ
C
u � dr
or
e1 �∇� u ¼ ∂ h3u
3ð Þ
∂x2
� ∂ h2u
2ð Þ
∂x3
198 4 Differential Operators
For the components u1 and u2 the result is, respectively,
e2 �∇� u ¼ ∂ h1u
1ð Þ
∂x3
� ∂ h3u
3ð Þ
∂x1
e3 �∇� u ¼ ∂ h2u
2ð Þ
∂x1
� ∂ h1u
1ð Þ
∂x2
Concluding that the curl components of the vectorial field u in the direction of
the upward unit normal vector to the closed surface S are given by
n �∇� u ¼ lim
S!0
1
S
þ
C
u � dr ð4:4:10Þ
This expression is valid for any type of referential system, which shows that the
curl is independent of the coordinate system.
For demonstrating that expression (4.4.9) represents the Stokes theorem, let the
surface S which outline is curve C, and the field represented by the vectorial
function u(r), continuous and with continuous partial derivatives in S [ C. Dividing
S in N cells Si, i ¼ 1, 2, . . .N, which components of the upward normal unit vectors
are ni, with closed outline curves Ci (Fig. 4.8a), and with expression (4.4.10) the
result for each cell of S is
ni �∇� u ¼ lim
Si!0
1
Si
þ
Ci
u � dr
Applying this expression to point P contained in cell Si with boundary Ci, the
result when the area of this cell is reduced approaching the outline P is
ni �∇� uð ÞSi ¼
þ
Ci
u � drþ h xi� �Si
where h xið Þk k > 0 is a function with very small value, which decreases with the
reduction of size of Si.
With the division of the surface S into N parts the result is thatN hi x
ið Þ½ � > 0, then
Max
1	i	N
hi x
ið Þ < h xið Þ. For N ! 1 the result is h xið Þ ! 0, so
ni �∇� uð ÞSi �
XN
i¼1
þ
Ci
u � dr
�������
������� < h xi
� �XN
i¼1
Si ¼ h xi
� �
S
4.4 Curl 199
and with
XN
i¼1
þ
Ci
u � dr ¼
þ
C
u � dr
for the outlines of the cells Si are calculated twice, but in opposite directions,
whereby these parcels cancel each other, leaving only the parcel of boundary
C of S. Then
ni �∇� uð ÞSi �
þ
C
u � dr
������
������ < h xi� �S
As N ! 1 the result is
lim
N!1
ni �∇� uð ÞSi ¼
þ
C
u � dr
whereby the result of the expression of Stokes theorem isðð
S
n �∇� u dS ¼
þ
C
u � dr
This theorem is a particular case of the divergence theorem. To demonstrate this
assertion let the vectorial function u ¼ v� w, and an arbitrary and constant vector,
then
∇ � v� wð Þ ¼ w �∇� v
and
n � v� wð Þ ¼ w � n� v
Applying the divergence theorem to the function u it is written asðð
S
u:ndS ¼
ððð
V
∇ � udV
The substitution of the previous expressions in this expression shows thatððð
V
∇ � udV ¼
ðð
S
v� wð Þ�n dS ¼
ðð
S
w � n� vð ÞdS
200 4 Differential Operators
whereby
w �
ððð
V
∇�v ¼ w �
ðð
S
n� vdS
As w is arbitrary it results inððð
V
∇�v ¼
ðð
S
n� v dS
The concept of curl of a vector u can be generalized for a space EN, in which the
vector is associated to an antisymmetric tensor A, and its order depends on the
dimension of the space. This tensor is generated by means of the dot product
between the Ricci pseudotensor and the vector’s covariant derivative
Ai1 i2���ip�2 ¼ εi1 i2���ip�2 j k∂juk ð4:4:11Þ
4.4.2 Orthogonal Curvilinear Coordinate Systems
With the expressions used in the previous item to demonstrate expression (4.4.10),
there is in index notation for the curl coordinates of vector u in a curvilinear
orthogonal coordinate system
∇� u ¼ hk
h1h2h3
∂hjuj
∂xi
� ∂hiu
i
∂xj
� �
gk ð4:4:12Þ
where hi ¼ ffiffiffiffiffiffiffiffig iið Þp , hj ¼ ffiffiffiffiffiffiffiffig jjð Þp , hk ¼ ffiffiffiffiffiffiffiffiffig kkð Þp are the components of the metric tensor,
and the indexes in parenthesis indicate no summation.
In a determinant form the result is
∇� u ¼ 1
h1h2h3
h1g1 h2g2 h3g3
∂
∂x1
∂
∂x2
∂
∂x3
h1u
1 h2u
2 h3u
3
							
							 ð4:4:13Þ
and with thephysical components of vector u*i it follows that
∇� u* ¼ hk
h1h2h3
∂hju*j
∂~xi
� ∂hiu
*i
∂xj
� �
gk ð4:4:14Þ
4.4 Curl 201
4.4.3 Properties
As the curl is the cross product of the nabla operator by a vectorial function, the
properties of this vector product are valid.
For the sum of two vectorial functions u and v:
∇� uþ uð Þ ¼ ∇� uþ∇� v ð4:4:15Þ
and the successive applying of the curl to this sum provides
∇�∇� uþ uð Þ ¼ ∇�∇� uþ∇�∇� v ð4:4:16Þ
Considering the vectorial function mu, where m is a scalar, the result of the cross
product
∇� muð Þ ¼ m∇� u ð4:4:17Þ
Expressions (4.4.16) and (4.4.17) show that the curl, for these cases, is a linear
operator, which is valid for the general case as will be shown in item 4.5.
4.4.4 Curl of a Tensor
The concept of curl of a vector in space EN is developed in an analogous way. For
instance, for the second-order tensor Tk1k2 in space EN exists the curl of order
p� 3ð Þ, given by the cross product between the Ricci pseudotensor of order p and
the tensor’s covariant derivative, then
Ai1 i2���ip�3 ¼ εi1 i2���ip�3 j k1k2∂jTk1k2 ð4:4:18Þ
Expression (4.4.18) shows that the Ricci pseudotensor is the generator of the
antisymmetric tensor that represents the rotational of the tensor.
In absolute notation the result is
∇� T ¼ ∇� Tð Þ�E ð4:4:19Þ
where E is the Ricci pseudotensor.
In the particular case of the space E4 the curl for the second-order tensor Tk‘ is
given by the components of vector Ai:
Ai ¼ εijk‘∂jTk‘
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Assuming that the second-order tensor is decomposed into two tensors, one
symmetric and the other antisymmetric
T ¼ Sþ A
then
rotT ¼ rotSþ rotA
The components of the curl of the symmetric tensor are given by
εi1 i2���ip�3 j k1k2∂jSk1k2 , i.e., are obtained by means of the dot product of the Ricci
pseudotensor (antisymmetric) by the symmetric tensor which is null, whereby
rotS ¼ 0. Concluding that the curl of a symmetric tensor is null, and that only the
antisymmetric tensor A generates the rotational of tensor T. In the particular case in
which rotT ¼ 0 the tensor T defines an irrotational tensorial field.
The definition of curl of a second-order tensor can be applied to a tensor of order
p > 2, whereby
Ai1 i2���iq�p�1 ¼ εi1 i2���iq�p�1 j k1���kp∂jTk1���kp ð4:4:20Þ
being q� 1ð Þ the order of the Ricci pseudotensor, and q� p� 1ð Þ the order of the
antisymmetric tensor that represents the curl of the tensor.
Exercise 4.12 Calculate: (a) ∇� ϕu; (b) u� ∇� vð Þ; (c) ∇� u� vð Þ.
(a) The curl of the field defined by the product of a scalar function ϕ(xi) for a
vectorial function u is given by
∇� ϕu ¼ g j � ∂ ϕukg
k
� �
∂xj
¼ g j � ∂ϕ
∂xj
ukg
k þ ϕ∂uk
∂xj
gk þ ϕuk ∂g
k
∂xj
� �
Substituting expression (2.4.4)
∂gk
∂xj
¼ �Γ kmjgm
in this expression
∇� ϕu ¼ g j � ∂ϕ
∂xj
ukg
k þ ϕ∂uk
∂xj
gk � ϕukΓ kmjgm
� �
The permutation of the indexes k $ m in the last term provides
∇� ϕu ¼ ∂ϕ
∂xj
uk þ ϕ∂uk∂xj � ϕumΓ
m
kj
� �
g j � gk
4.4 Curl 203
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and with expressions
g j � gk ¼ εijkgi ∂juk ¼
∂uk
∂xj
� umΓmkj
it follows that
∇� ϕu ¼ ∂ϕ
∂xj
uk þ ϕ∂juk
� �
εijkgi ¼
∂ϕ
∂xj
ukε
ijkgi þ ϕ ∂juk
� �
εijkgi
Putting ∂juk ¼ ∇uk
∇� ϕu ¼ ∇ϕ� uþ ϕ∇� u ) rotϕu ¼ gradϕ� uþ ϕ rotu
For the Cartesian coordinates
∇� ϕu ¼ εijk uk ∂ϕ∂xj þ εijkϕ
∂uk
∂xj
� �
gi
It is verified for this case that the curl is not a linear operator. Forϕ ¼ m, where
m is a constant, the result with expression (4.4.17) is that this operator’s
linearity is valid by this particular case.
(b) The curl ∇� v is given by
∇� v ¼ w ¼ εkmn∂mvngk
then
u� w ¼ εijku jwkgi
whereby substituting
u� ∇� vð Þ ¼ εijku jεkmn∂mvngi ¼ εijkεkmnu j∂mvngi
and with
εijkε
kmn ¼ δmnij δmnij ¼ δmi δ nj � δmj δni
is it follows that
u� ∇� vð Þ ¼ δmi δnj u j∂mvn � δmj δ ni u j∂mvn
h i
gi ¼ u j∂ivj � u j∂jvi
� �
gi
204 4 Differential Operators
For the Cartesian coordinates the result is
u� ∇� vð Þ ¼ uj ∂vj∂xi � uj
∂vi
∂xj
� �
gi
(c) The cross product u� v ¼ w is given by
u� v ¼ w ¼ w‘g‘ ¼ ε‘mnumvng‘
thereby
∇� u� vð Þ ¼ εij‘∂jw‘gi ¼ εij‘∂j ε‘mnumvnð Þgi ¼ δijmn∂j umvnð Þgi
¼ δ imδ jn � δ inδ jm
� �
∂j umvnð Þgi ¼ ∂j uiv jð Þ � ∂j u jvið Þ
� �
gi
For the Cartesian coordinates
∇� u� vð Þ ¼ vj ∂u
i
∂xj
þ ui ∂v
j
∂xj
� vi ∂u
j
∂xj
� uj ∂v
i
∂xj
� �
gi ¼ v �∇� u� u �∇� v
Exercise 4.13 Calculate ∇� u for the vector u expressed in cylindrical
coordinates.
For the cylindrical coordinates the result is h1 ¼ 1, h2 ¼ r, h3 ¼ 1 and (ur, uθ, uz).
The determinant is given by the expression (4.4.13)
∇� u ¼ 1
r
gr rgθ gz
∂
∂xr
∂
∂xθ
∂
∂xz
ur ruθ uz
								
								
which development provides
∇� u ¼ 1
r
∂uz
∂θ
� ∂uθ
∂z
� �
gr þ
∂ur
∂z
� ∂uz
∂r
� �
gθ þ
1
r
∂ruθ
∂r
� 1
r
∂ur
∂θ
� �
gz
Exercise 4.14 Calculate∇� u for the vector u expressed in spherical coordinates.
For the spherical coordinates the result is h1 ¼ 1, h2 ¼ r, h3 ¼ r sin ϕ and
(ur, uϕ, uθ). The determinant given by expression (4.4.13)
4.4 Curl 205
∇� u ¼ 1
r2 sin ϕ
gr rgκ r sin ϕgθ
∂
∂xr
∂
∂xϕ
∂
∂xθ
ur ruϕ r sin ϕuθ
									
									
which development provides
∇� u ¼ 1
r sin ϕ
∂uθ sin ϕ
∂ϕ
� ∂uϕ
∂θ
� �
gr þ
1
r sin ϕ
∂uθr sin ϕ
∂r
� 1
r
∂ur
∂θ
� �
gϕ þ
1
r
∂ruϕ
∂r
� ∂ur
∂ϕ
� �
gθ
Exercise 4.15 Let r be the position vector of the point in space E3, show that: (a)
∇� r ¼ 0; (b) ϕ(r)r is irrotational.
(a) With the definition of curl
∇� r ¼ gi
∂
∂xi
� �
� r ¼ gi �
∂r
∂xi
but
∂r
∂xi
¼ gi
then
∇ � r ¼ gi � gi ¼ 0 Q:E:D:
(b) A condition that a vectorial function must fulfill so that the field that it
represents is irrotational is
∇� ϕ rð Þr½ � ¼ 0
and putting
ϕ rð Þ ¼ ψ
it follows that
∇� ϕ rð Þr½ � ¼ gradϕ� rþ ϕ∇� r ¼ ϕ0 rð Þgrad r
h i
� rþ ϕ rð Þ �∇� r
206 4 Differential Operators
but
∇� r ¼ 0
then
∇� ϕ rð Þ r½ � ¼ ϕ0 rð Þ1
r
r
 �
� r
as
r� r ¼ 0
thus
∇� ϕ rð Þ r½ � ¼ 0 Q:E:D:
4.5 Successive Applications of the Nabla Operator
The operator∇ can be applied successively to a field. The number of combinations
of two out of the three differential operators, the gradient, the divergence, and
the curl, are 32 ¼ 9 types of double operators. The combinations ∇ � ∇ � uð Þ and
∇� ∇ � uð Þ have no mathematical meaning.
4.5.1 Basic Relations
(1) ∇ � ∇� uð Þ
The curl of a vectorial function is given by
∇� u ¼ εk‘m∂‘umgk ¼ w ¼ wkgk ð4:5:1Þ
wk ¼ εk‘m∂‘um ð4:5:2Þ
4.5 Successive Applications of the Nabla Operator 207
then
∇ � ∇� uð Þ ¼ gi � ∂ w
kgk
� �
∂xi
¼ gi � ∂w
k
∂xi
gk þ wk
∂gk
∂xi
� �
and with expression
∂gk
∂xi
¼ Γ nkign
it follows that
∇ � ∇� uð Þ ¼ gi � ∂w
k
∂xi
gk þ wkΓ nkign
� �
The permutation of indexes n $ k in the second member in parenthesis
provides
∇ � ∇� uð Þ ¼ ∂w
k
∂xi
þ wnΓ kni
� �
gi � gk
and with
gi � gk ¼ δ ik
the result is
∇ � ∇� uð Þ ¼ ∂w
k
∂xk
þ wnΓ knk
� �
¼ ∂kwk
Substituting expression (4.5.2)
∇ � ∇� uð Þ ¼ ∂k εk‘m∂‘um
� � ¼ εk‘m∂k ∂‘umð Þ ¼ ek‘mffiffiffi
g
p ∂k ∂‘umð Þ
and interchanging the indexes i, j, k ¼ 1, 2, 3 cyclically
∇ � ∇� uð Þ ¼ 1ffiffiffi
g
p ∂k∂‘um � ∂‘∂kumð Þ ð4:5:3Þ
whereby
∇ � ∇� uð Þ ¼ 0 ð4:5:4Þ
208 4 Differential Operators
Vector∇� u represents a vectorial field associated to the vectorial function u.
Expression (4.5.4) defines the condition of existence for this function. The
property of the field defined by the curl of the vectorial function u shows that
the divergence of this field is null, i.e., the field is solenoidal.
In a reciprocal way for a solenoidal field ∇ � ∇� uð Þ ¼ 0 a solenoidal
vector v can be determined, such that v ¼ ∇� uð Þ. In this case the vector v
derives from the potential function u, being linked to this function.
(2) ∇� ∇ϕð Þ
The gradient of a scalar function ϕ(xi) is given by
∇ϕð Þ ¼ u ¼ ∂ϕ
∂xk
gk ¼ ukgk ð4:5:5Þ
then
∇� ∇ϕð Þ ¼ ∇u ¼ εijk∂jukgi
it follows that
∇� ∇ϕð Þ ¼ εijk∂j ∂ϕ∂xk
� �
gi ¼ εijk
∂2ϕ
∂xj∂xk
 !
gi ¼
eijkffiffiffi
g
p ∂
2ϕ
∂xj∂xk
 !
gi
Interchangingthe indexes i, j, k ¼ 1, 2, 3 cyclically
∇� ∇ϕð Þ ¼ 1ffiffiffi
g
p ∂
2ϕ
∂xj∂xk
� ∂
2ϕ
∂xk∂xj
 !
gi
As
∂2ϕ
∂xj∂xk
¼ ∂
2ϕ
∂xk∂xj
it results in
∇� ∇ϕð Þ ¼ 0 ð4:5:6Þ
The field that fulfills the condition given by expression (4.5.6) is called a
conservative field, i.e., every vectorial field with potential is an irrotational
field.
Let ∇ϕ ¼ u the result is ∇� ∇ϕð Þ ¼ ∇� u ¼ 0, then
4.5 Successive Applications of the Nabla Operator 209
∂ui
∂xj
¼ ∂uj
∂xi
As uidx
i is an exact differential it follows that for a scalar function ϕ(xi):
ϕ xi
� � ¼ ∂ϕ
∂xi
dxi ) ui � ∂ϕ∂xi
� �
dxi ¼ 0
whereby
ui ¼ ∂ϕ∂xi
This analysis shows that the vector u can be considered as the gradient of a
scalar function ϕ(xi) as long as it fulfills the condition ∇� u ¼ 0.
Expression (4.5.6) can be demonstrated changing only the order of the opera-
tions, for ∇�∇ð Þϕ ¼ 0, where the term in parenthesis indicates the cross
product of a vector by itself, which results in the null vector.
The condition ∇� u ¼ 0 being u ¼ ∇ϕ xið Þ, where the scalar field defined by
the function ϕ(xi) is divided into families of level surfaces ϕ xið Þ ¼ C, which do
not intersect, so they form level surface “layers,” leads to the denomination of
lamellar field.
(3) ∇� ∇� uð Þ
For ∇� ∇� uð Þ using the Grassmann identity
u� v� wð Þ ¼ u � wð Þv� u � vð Þw
whereby
∇� ∇� uð Þ ¼ ∇ ∇ � uð Þ �∇ � ∇uð Þ ð4:5:7Þ
In terms of the vector coordinates it follows that
∇� u ¼ εtjk ∂‘uk
� �
g‘jgt ¼ w ¼ wtgt
wt ¼ εtjk ∂‘uk
� �
g‘j
∇� w ¼ gs � ∂ wtg
tð Þ
∂xs
¼ gs � ∂wt
∂xs
gt þ wt ∂g
t
∂xs
� �
∂gt
∂xs
¼ �Γ tsngn
210 4 Differential Operators
∇� w ¼ gs � ∂wt
∂xs
gt � wtΓ tsngn
� �
¼ ∂wt
∂xs
� wnΓ nst
� �
gs � gt
gs � gt ¼ εrstgr
∇� w ¼ εrst ∂wt
∂xs
� wnΓ nst
� �
gr
∂sε
rst ¼ 0
∇� w ¼ εrst ∂swtð Þgr ¼ εrstεijk ∂s∂‘ukg‘j
� �
gr
∂sg
‘j ¼ 0
∇� w ¼ εrstεtjk ∂s∂‘uk
� �
g‘j gr
and with
εrstεtjk ¼ δrsjk δrsjk ¼ δ rj δ sk � δ sj δ rk
it follows that
∇� ∇� uð Þ ¼ δ rj δ sk ∂s∂‘uk
� �
g‘j � δ sj δ rk ∂s∂‘uk
� �
g‘j
h i
gr
whereby
∇� ∇� uð Þ ¼ ∂k∂‘uk
� �
g‘r � ∂j∂‘ur
� �
g‘j
� �
gr ð4:5:8Þ
∇� ∇� uð Þ ¼ ∂k∂ruk
� �
g‘r � ∂j∂jur
� �h i
gr ð4:5:9Þ
For the Cartesian coordinates the result is
∇� ∇� uð Þ ¼ ∂
2
uk
∂xk∂xr
� ∂
2
ur
∂2xj
 !
gr ð4:5:10Þ
(4) ∇ ∇ � uð Þ
For the gradient of a vector
∇ � u ¼ ∂u
i
∂xi
þ ujΓ iji
� �
4.5 Successive Applications of the Nabla Operator 211
then
∇ ∇ � uð Þ ¼ gm ∂
∂xm
∂ui
∂xi
þ ujΓ iji
� �
The development provides
∇ ∇ � uð Þ ¼ gm ∂
2
ui
∂xm∂xi
þ ∂u
j
∂xm
Γ iji þ uj
∂Γ iji
∂xm
 !
or
∇ ∇ � uð Þ ¼ ∂m ∂iui
� �
gm ð4:5:11Þ
and with
u ¼ ϕ0 rð Þ
du
dr
¼ ϕ0 rð Þ
it follows that
∇ ∇ � uð Þ ¼ gmk∂m ∂iui
� �
gk
whereby
∇ ∇ � uð Þ ¼ ∂k ∂iui
� �
gk ð4:5:12Þ
Exercise 4.16 Let ϕ(xi) and ψ(xi) be scalar functions, show that: (a)
∇� ψ∇ϕþ ϕ∇ψð Þ ¼ 0; (b) ∇ � ∇ϕ�∇ψð Þ ¼ 0; (c) tr ∇� uð Þ ¼ ∇ � u.
(a) Putting
∇ϕ ¼ u ∇ψ ¼ v
then
∇� ψ∇ϕþ ϕ∇ψð Þ ¼ ∇� ψuþ ϕvð Þ ¼ ∇� ψuþ∇� ϕv
and with the expression shown in Exercise 4.12 it follows that
∇� ψu ¼ ∇ψ � uþ ψ �∇u ¼ ∇ψ �∇ϕþ ψ∇�∇ϕ
∇� ϕv ¼ ∇ϕ� vþ ϕ�∇v ¼ ∇ϕ�∇ψ þ ϕ∇�∇ψ
212 4 Differential Operators
and with expression (4.5.6)
∇�∇ϕ ¼ ∇�∇ψ ¼ 0
and
∇ϕ�∇ψ ¼ �∇ψ �∇ϕ
then
∇� ψ∇ϕþ ϕ∇ψð Þ ¼ 0 Q:E:D:
(b) Putting
∇ϕ ¼ u ∇ψ ¼ v ∇� ∇ϕ�∇ψð Þ ¼ 0
then
∇ � ∇ϕ�∇ψð Þ ¼ ∇ � u� vð Þ
With expression deducted in Exercise 4.7b it follows that
∇ � u� vð Þ ¼ v �∇� u� u �∇� v
∇� ∇ϕ�∇ψð Þ ¼ 0
and with expression (4.5.6)
∇�∇ϕ ¼ ∇�∇ψ ¼ 0
then
∇ � ∇ϕ�∇ψð Þ ¼ 0 Q:E:D:
(c) With expression (4.2.11)
∇� u ¼ ∂iukð Þgi � gk
the result for i ¼ k is
tr ∇� uð Þ ¼ ∂iui1
and comparing this result with expression (4.3.6)
4.5 Successive Applications of the Nabla Operator 213
∇ � u ¼ ∂iui
it is verified that
tr ∇� uð Þ ¼ ∇ � u Q:E:D:
4.5.2 Laplace Operator
The combination of the divergence and the gradient, in this order, defines the
Laplace operator or Laplacian
∇2 ¼ ∇ �∇ ¼ Δ ¼ Dk � Dk ¼ ∂k∂k ¼ divgrad ¼ lap ð4:5:13Þ
A few authors denominate this operator of differential parameter of the second
order of Beltrami, and use the spelling Δ 2 to represent it.
With the expression the contravariant derivative
∂k ¼ gkj∂j
it follows that for the Laplacian of an arbitrary tensor
∂k∂
k
T������ ¼ ∂k ∂kT������
� �
¼ ∂k gkj∂jT������
� � ¼ gkj∂k ∂jT������� � ¼ ∂j∂jT ������
that shows that the Laplacian operator is commutative.
For Cartesian coordinates the covariant and contravariant derivatives are equal
∂k ¼ ∂� � �∂xk ∂
k ¼ ∂� � �
∂xk
∂k ¼ ∂k
resulting for the Laplacian ð
dϕ rð Þ ¼
ð
m1
r2
dr
4.5.2.1 Laplacian of a Scalar Function
The Laplacian of the scalar function ϕ(xi) expresses in a curvilinear coordinate
system, with covariant derivative given by
214 4 Differential Operators
ϕ rð Þ ¼ m1
r
þ m2
thus
H � � �ð Þ ¼ gi∇� gj∇ � � �ð Þ
The development of the covariant derivative of the term in parenthesis provides
ϕ xi
� �
The contracted Christoffel symbol
Γ kmk ¼
1ffiffiffi
g
p ∂
ffiffiffi
g
p� �
∂xm
provides
∇2ϕ ¼ ∂
∂xk
gkj
∂ϕ
∂xj
� �
þ gmj ∂ϕ
∂xj
1ffiffiffi
g
p ∂
ffiffiffi
g
p� �
∂xm
whereby
H ϕð Þ ¼ ∂
∂x1
∂
∂x2
∂
∂x3

 �
gi � gj ∂
∂x1
∂
∂x2
∂
∂x3

 �
ϕ
¼
∂2ϕ
∂x1∂x1
∂2ϕ
∂x1∂x2
∂2ϕ
∂x1∂x3
∂2ϕ
∂x2∂x1
∂2ϕ
∂x2∂x2
∂2ϕ
∂x2∂x3
∂2ϕ
∂x3∂x1
∂2ϕ
∂x3∂x2
∂2ϕ
∂x3∂x3
2666666664
3777777775
gi � gj
or
∇2ϕ ¼ gik ∂jkϕ
� � ð4:5:14Þ
it follows that
∇2ϕ ¼ gik ∂
2ϕ
∂xj∂xk
� ∂ϕ
∂xm
Γmjk
 !
ð4:5:15Þ
4.5 Successive Applications of the Nabla Operator 215
In vectorial notation
∇ � ∇ϕð Þ ¼ ∇2ϕ ð4:5:16Þ
or
div gradϕð Þ ¼ ∇2ϕ ð4:5:17Þ
In space E3 and in orthogonal Cartesian coordinates the result is gij ¼ δij, then
the Laplacian of a scalar function is the sum of its derivatives of the second order
∇2ϕ ¼ ∂
2ϕ
∂xj∂xj
ð4:5:18Þ
4.5.3 Properties
The Laplacian of the sum of two scalar functions ϕ(xi) and ψ(xi) provides
∇2 ϕþ ψð Þ ¼ ∇ �∇ ϕþ ψð Þ ¼ ∇ � ∇ϕþ∇ψð Þ ¼ ∇ �∇ϕþ∇ �∇ψ
whereby
∇2 ϕþ ψð Þ ¼ ∇2ϕþ∇2ψ
For the function mϕ(xi), where m is a scalar, this operator provides
∇2 mϕð Þ ¼ ∇ �∇ mϕð Þ ¼ ∇ � m∇ ϕð Þ ¼ m∇ �∇ ϕð Þ
whereby
∇2 mϕð Þ ¼ m∇ �∇ ϕð Þ
These two demonstrations prove that the Laplacian is a linear operator.
The gradient of the product of two scalar functions is given by
∇ ϕψð Þ ¼ ψ∇ ϕð Þ þ ϕ∇ ψð Þ
then
∇ �∇ ϕψð Þ ¼ ∇ � ψ∇ ϕð Þ þ ϕ∇ ψð Þ½ �
216 4 Differential Operators
Putting
∇ϕ ¼ u ∇ψ ¼ v
thus
∇ � ψ∇ ϕð Þ þ ϕ∇ ψð Þ½ � ¼ ∇ � ψuþ ϕvð Þ
Applying the distributive property of the divergence to this expression, and using
the expression deducted in Exercise 4.7a it follows that
∇ � ψuþ ϕvð Þ ¼ ∇ � ψuþ∇ � ϕv
∇ � ψu ¼ ∇ψ � uþ ψ∇u
∇ � ϕv ¼ ∇ϕ � vþ ϕ∇v
then
∇2 ϕψð Þ ¼ ∇ψ � uþ ψ∇uð Þ þ ∇ϕ � vþ ϕ∇vð Þ
¼ v � uþ ψ∇uð Þ þ u � vþ ϕ∇vð Þ
¼ ψ∇uþ ϕ∇vþ 2 v � uð Þ
but
∇u ¼ ∇∇ϕ ¼ ∇2ϕ ∇v ¼ ∇∇ψ ¼ ∇2ψ
whereby substituting
∇2 ϕψð Þ ¼ ψ∇2 ϕð Þ þ ϕ∇2 ψð Þ þ 2∇ϕ∇ψ
An equation involving the Laplacian of a scalar function that appears in various
problems of physics and engineering, the Laplace equation, is given by
∇2ϕ xi
� � ¼ 0 ð4:5:19Þ
The functionϕ xið Þ ¼ x4zþ 3xy2 � zxyþ 1 that fulfills this equation is said to be
harmonic. In addition to satisfying the Laplace equation it must be regular in the
domain D, with partial derivatives of the first order continuous in the interior and in
the boundary of D, and derivatives of the second order also continuous in D, which
can be discontinuous in the boundary of this domain.
The successive applying of the Laplacian to a scalar function (r; z sin θ; eθ cos z )
results in the bi-harmonic equation
ϕ xi
� � ¼ xyþ yzþ xz ð4:5:20Þ
4.5 Successive Applications of the Nabla Operator 217
For
∂� � �
∂t2
ð4:5:21Þ
where ψ(xi) is a scalar function, this partial differential equation is called Poisson’s
equation.
As a consequence of the definition of the Laplacian the result is that ∇2m ¼ 0,
where m is a scalar. The Laplacian of a scalar function ϕ(xi) is a scalar, then its
physical components are equal to its ordinary components.
4.5.4 Orthogonal Coordinate Systems
With the gradient of the scalar function ϕ(xi):
∇ϕ xi
� � ¼ gi ∂ϕ
∂xi
¼ u
and the orthogonalcomponents of the vectorial function u given by
∇ � u ¼ 1
h1h2h3
∂
∂xi
uih1h2h3
hi
� �
results for the Laplacian of this function expressed in an orthogonal coordinate
system
∇2ϕ ¼ 1
h1h2h3
∂
∂xi
h1h2h3
hi
∂ϕ
∂xi
� �
ð4:5:22Þ
where h1, h2, h3 are the components of the metric tensor.
4.5.5 Laplacian of a Vector
With expression (4.5.7)
∇2u ¼ ∇ ∇ � uð Þ �∇� ∇� uð Þ
and substituting expressions (4.5.12) and (4.5.9) this expression becomes
218 4 Differential Operators
∇2u ¼ gk∂k ∂iui
� �� ∂k∂ruk� �� ∂j∂jur� �h igr ð4:5:23Þ
The change of the indexes k ! r in the first term to the right and indexes k ! i in
the second term to the right provides
∇2u ¼ ∂r∂iui
� �� ∂i∂rui� �þ ∂j∂jur� �h igr
As
∂r∂iu
i ¼ ∂i∂rui
thus
∇2u ¼ ∂j∂jur
� �
gr ð4:5:24Þ
4.5.6 Curl of the Laplacian of a Vector
The curl of the Laplacian of a vector is∇�∇2u, and it can be developed by means
of the Grassmann formula
∇2u ¼ divgradu ¼ ∇∇ � u�∇�∇� u ð4:5:25Þ
or
∇�∇� u ¼ ∇∇ � u�∇2u
The curl of this expression is given by
∇�∇�∇� u ¼ ∇�∇∇ � u�∇�∇2u ð4:5:26Þ
or
∇�∇�∇� u ¼ ∇∇ � ∇� uð Þ �∇ �∇ ∇� uð Þ ð4:5:27Þ
Expressions (4.5.4) and (4.5.6) show, respectively, that
∇ � ∇� uð Þ ¼ 0
∇�∇ϕ ¼ 0
4.5 Successive Applications of the Nabla Operator 219
whereby the result for expression (4.5.26) is
∇�∇�∇� u ¼ ∇�∇∇ � u�∇�∇2u ¼ ∇�∇ϕ�∇�∇2u
¼ �∇�∇2u
and for expression (4.5.27)
∇�∇�∇� u ¼ �∇ �∇ ∇� uð Þ
The result of these two expressions is
∇�∇2u ¼ ∇2 ∇� uð Þ ð4:5:28Þ
or
rot lapu ¼ lap rotu ð4:5:29Þ
It is concluded that the operators ∇2 and∇� are commutative when applied to
vector u.
4.5.7 Laplacian of a Second-Order Tensor
The gradient of a second-order tensor is given by
∇� T ¼ ∂mTijgm � gi � gp
and the divergence of the tensor defined by the previous expression stays
∇ �∇� T ¼ gk � ∂k∂mTijgm � gi � gp ¼ ∂k∂mTijgm � gi � gp � gk
¼ ∂k∂mTijgm � giδ kp
whereby
∇ �∇� T ¼ ∂p∂mTijgm � gi ð4:5:30Þ
is a second-order tensor.
Exercise 4.17 Calculate∇2ϕ for the scalar function ϕ(xi) expressed in cylindrical
coordinates.
The tensorial expression that defines the Laplacian is
220 4 Differential Operators
∇2ϕ ¼ 1ffiffiffi
g
p ∂
∂xk
ffiffiffi
g
p
gkj
∂ϕ
∂xj
� �
and for the cylindrical coordinates
g11 ¼ 1 g22 ¼ 1
r2
g33 ¼ 1
∇ϕ ¼ ∂ϕ
∂r
gr þ
1
r
∂ϕ
∂θ
gθ þ
∂ϕ
∂z
gz
it follows that
∇2ϕ ¼ 1
r
∂
∂r
r
∂ϕ
∂r
� �
þ ∂
∂θ
1
r
∂ϕ
∂ϕ
� �
þ ∂
∂z
r
∂ϕ
∂z
� �
 �
then
∇2ϕ ¼ 1
r
∂ϕ
∂r
þ ∂
2ϕ
∂r2
þ 1
r2
∂2ϕ
∂θ2
þ ∂
2ϕ
∂z2
Exercise 4.18 Calculate ∇2ϕ for the scalar function ϕ(xi) expressed in spherical
coordinates.
The tensorial expression that defines the Laplacian is
∇2ϕ ¼ 1ffiffiffi
g
p ∂
∂xk
ffiffiffi
g
p
gkj
∂ϕ
∂xj
� �
and for the spherical coordinates
g11 ¼ 1 g22 ¼ 1
r2
g33 ¼ 1
r2 sin 2ϕ
∇ϕ ¼ ∂ϕ
∂r
gr þ
1
r
∂ϕ
∂ϕ
gϕ þ
1
r sin ϕ
∂ϕ
∂xθ
gθ
it follows that
∇2ϕ¼ 1
r2 sinϕ
∂
∂r
r2 sinϕ
∂ϕ
∂r
� �
þ ∂
∂ϕ
r2 sinϕ
1
r2
∂ϕ
∂ϕ
� �
þ ∂
∂θ
r2 sinϕ
1
r2 sin2ϕ
∂ϕ
∂θ
� �
 �
then
4.5 Successive Applications of the Nabla Operator 221
∇2ϕ ¼ 2
r
∂ϕ
∂r
þ ∂
2ϕ
∂r2
þ 1
r2 sin ϕ
∂
∂ϕ
sin ϕ
∂ϕ
∂ϕ
� �
þ 1
r2 sin 2ϕ
∂2ϕ
∂θ2
Exercise 4.19 Let r be the position vector of the point in space E3, show that: (a)
∇2 xr3
� � ¼ 0; (b)∇2 rnrð Þ ¼ n nþ 3ð Þ rn�2r; (c)∇2ϕ rð Þ ¼ ϕ00 rð Þ þ 2r ϕ0 rð Þ; (d) for∇2
ϕ rð Þ ¼ 0 the result is ϕ rð Þ ¼ m1r þ m2, where m1,m2 are constant.
(a) With the definition of Laplacian
∇2
x
r3
� �
¼ ∂
2
∂x2
þ ∂
2
∂y2
þ ∂
2
∂z2
 !
x
r3
� �
and for the derivative with respect to the variable x
∂2
∂x2
x
r3
� �
¼ ∂
∂x
∂
∂x
x
r3
� �
 �
¼ ∂
∂x
1
r3
� 3x
r4
∂r
∂x
 �
but
2r
∂r
∂x
¼ 2x
then
∂2
∂x2
x
r3
� �
¼ ∂
∂x
1
r3
� 3x
r4
x
r
 �
¼ � 3
r4
∂r
∂x
� 6x
r5
þ 15x
2
r6
∂r
∂x
¼ � 9x
r5
þ 15x
3
r2
In an analogous way for the other derivatives it follows that
∂2
∂y2
y
r3
� �
¼ � 3x
r5
þ 15xy
r7
∂2
∂z2
y
r3
� �
¼ � 3x
r5
þ 15xz
r7
Then
∇2
x
r3
� �
¼ � 9x
r5
þ 15x
3
r2
� 3x
r5
þ 15xy
r7
� 3x
r5
þ 15xz
r7
∇2
x
r3
� �
¼ 0 Q:E:D:
(b) Putting
∇2 rnrð Þ ¼ ∇ ∇ � rnrð Þ½ �
222 4 Differential Operators
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it follows that
∇2 rnrð Þ ¼ ∇ ∇ rnð Þ � rþ rn∇ � r½ � ¼ ∇ nrn�3r� � � rþ 3rn� �
¼ ∇ nrn�3r2� �þ 3rn� � ¼ nþ 3ð Þ∇rn
then
∇2 rnrð Þ ¼ nþ 3ð Þnr�2r Q:E:D:
(c) With the definition of Laplacian
∇2ϕ rð Þ ¼ ∇ � ∇ϕ rð Þ½ �
it follows that
∇ � ∇ϕ rð Þ½ � ¼ ∇ � ϕ0 rð Þ∇r
h i
¼ ∇ � 1
r
ϕ
0
rð Þr
 �
¼ 1
r
ϕ
0
rð Þ∇ � rþ r �∇ 1
r
ϕ
0
rð Þ
 �
but
∇ � r ¼ 3
so
∇ � ∇ϕ rð Þ½ � ¼ 3
r
ϕ
0
rð Þ∇ � rþ r � d
dr
1
r
ϕ
0
rð Þ
 �
∇r

 �
¼ 3
r
ϕ
0
rð Þ þ r � � 1
r2
ϕ
0
rð Þ þ 1
r
ϕ
00
rð Þ
 �
1
r
r

 �
¼ 3
r
ϕ
0
rð Þ þ � 1
r2
ϕ
0
rð Þ þ 1
r
ϕ
00
rð Þ
 �
1
r

 �
r � r
¼ 3
r
ϕ
0
rð Þ þ � 1
r2
ϕ
0
rð Þ þ 1
r
ϕ
00
rð Þ
 �
1
r

 �
r2
then
∇ � ∇ϕ rð Þ½ � ¼ 2
r
ϕ
0
rð Þ þ ϕ00 rð Þ
(d) Let
∇2ϕ rð Þ ¼ ϕ00 rð Þ þ 2
r
ϕ
0
rð Þ ) ϕ00 rð Þ þ 2
r
ϕ
0
rð Þ ¼ 0 ) ϕ
00
rð Þ
ϕ
0
rð Þ ¼ �
2
r
4.5 Successive Applications of the Nabla Operator 223
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Putting
u ¼ ϕ0 rð Þ ) du
dr
¼ ϕ0 rð Þ
then
du
u
¼ �2
r
dr
and integrating ð
du
u
¼ �
ð
2
r
dr
it follows that
‘n uð Þ ¼ �‘n r2� �þ ‘n m1ð Þ ¼ ‘n m1
r2
� �
or
‘nϕ
0
rð Þ ¼ ‘n m1
r2
� �
) ϕ0 rð Þ ¼ m1
r2
Integrating ð
dϕ rð Þ ¼
ð
m1
r2
dr
then
ϕ rð Þ ¼ m1
r
þ m2 Q:E:D:
4.6 Other Differential Operators
4.6.1 Hesse Operator
The operator defined on a scalar field, given by the tensorial product of two nabla
operators applied to the scalar function that field represents
224 4 Differential Operators
H � � �ð Þ ¼ gi∇� gj∇ � � �ð Þ ð4:5:31Þ
In matrix form the scalar function ϕ(xi) in Cartesian coordinates in the space E3
is
H ϕð Þ ¼ ∂
∂x1
∂
∂x2
∂
∂x3

 �
gi � gj ∂
∂x1
∂
∂x2
∂
∂x3

 �
ϕ
¼
∂2ϕ
∂x1∂x1
∂2ϕ
∂x1∂x2
∂2ϕ
∂x1∂x3
∂2ϕ
∂x2∂x1
∂2ϕ
∂x2∂x2
∂2ϕ
∂x2∂x3
∂2ϕ
∂x3∂x1
∂2ϕ
∂x3∂x2
∂2ϕ
∂x3∂x3
266666664
377777775
gi � g j ð4:5:32Þ
This operator is a symmetric second-order tensor, is called Hessian or Hesse
operator in homage to Ludwig Otto Hesse (1881–1874).
4.6.2 D’Alembert Operator
The differential operator defined by the expression
□ ¼ ∇2� � � þ 1
c2
∂� � �
∂t2
ð4:5:33Þ
where c is a scalar and
∂� � �
∂t2
denotes the differentiation with respect to the time t, is
called D’Alembert or D’Alembertian operator in homage to Jean Le Rond
d’Alembert (1717–1783).
The applying of this operator to a field represented by the scalar function that
depends on the position vector and the time provides as a result the scalar function
□ ϕ xi; t
� � ¼ ∇2ϕ xi; t� �þ 1
c2
∂ϕ xi; tð Þ
∂t2
ð4:5:34Þ
If the field is represented by a vectorial function the result is the vector
□ u xi; t
� � ¼ ∇2u xi; t� �þ 1
c2
∂u xi; tð Þ
∂t2
ð4:5:35Þ
The notation □. . . was initially applied by Cauchy to represent the Laplacian.
The D’Alembertian is the four-dimensional equivalent to the Laplacian.
4.6 Other Differential Operators 225
Problems
4.1 Calculate the gradient of the scalar functions:
(a) ϕ xið Þ ¼ xyþ yzþ xz; (b) ϕ xið Þ ¼ xex2þy2 .
Answer: (a) yþ zð Þiþ xþ zð Þjþ xþ yð Þk; (b) 1þ 2x2ð Þex2þy2 iþ 2xyex2þy2 j
4.2 Calculate the directional derivative of the scalar function ϕ ¼ 2 x1ð Þ2 þ 3 x2ð Þ 2
þ x3ð Þ2 at the point (2; 1; 3) in the direction of vector u 1; 0;�2ð Þ.
Answer: �1:789.
4.3 Calculate div ui with the vector u expressed in cylindrical coordinates by its
covariant components (r; z sin θ; eθ cos z).
Answer: divui ¼ 2þ 1rð Þ 2 z cos θ � eθ sin z
4.4 Show that∇ � ϕψuð Þ ¼ ϕ∇ψ � uþ ψ∇ϕ � uþ ϕψ∇ � u, where ϕ,ψ are scalar
functions and u is a vectorial function.
4.5 Calculate the curl of the following vectorial fields:
(a) y2iþ z2jþ x2k; (b) xyz xiþ yjþ zkð Þ.
Answer: (a) �2 ziþ xjþ ykð Þ; (b) xz2 � xy2ð Þ iþ x2y� yz2ð Þ jþ y2z� x2zð Þk.
4.6 Calculate the Laplacian of the function ϕ xið Þ ¼ x4zþ 3xy2 � zxyþ 1.
Answer: ∇2ϕ ¼ 12x2z� 6x.
226 4 Differential Operators
Chapter 5
Riemann Spaces
5.1 Preview
The space provided with metric is called Riemann space, for which the tensorial
formalism is based on the study inits first fundamental form, being complemented
by the definition of curvature and by the concept of geodesics, which allows
expanding the basic conceptions of the Euclidian geometry for this type of space
with N dimensions.
In the Riemann spaces the covariant derivatives of tensors are equal to the partial
derivatives when the coordinates are Cartesian, but the problem arises of
researching how these derivatives behave when the coordinate system is curvilin-
ear. The analysis of this derivative leads to the definition of curvature of the space,
which is the fundamental parameter for the development of a consistent study of the
Riemann spaces EN.
The concepts and expressions of Tensor Calculus are essential for the formula-
tion of the Theory of General Relativity, and it is for this theory just as the Integral
and Differential Calculus is for the Classic Mechanics.
5.2 The Curvature Tensor
The Euclidian geometry is grounded on the basic concepts of point, straight line,
and plane, and in various axioms. In this geometry a curved line is defined in the
Euclidian space E2 as the one that is not a straight line, and in the Euclidian space E3
a curved surface is defined as the one that is not a plane. The curvature is an intrinsic
characteristic of the space, so it is not a property measurable by comparison
between distinct spaces.
© Springer International Publishing Switzerland 2016
E. de Souza Sánchez Filho, Tensor Calculus for Engineers and Physicists,
DOI 10.1007/978-3-319-31520-1_5
227
The conception of a Riemann geometry for the space EN is grounded on the basic
concepts of the Euclidian geometry in space E3, which generalization is carried out
by means of defining the metric for the space EN, given by
ds2 ¼ εgijdxidxj
where ε ¼ 
1 is a functional indicator.
The space in which metric can be writed as an Euclidian metric, positive and
definite, is called a flat space, otherwise it is called space with curvature. The
concept of curvature of the space EN was firstly conceived by Riemann as a
generalization of the study of a surface’s curvature developed by Gauß. Riemann
presented his results in a paper in 1861, published only in 1876. Christoffel in 1869
and R. Lipschitz in four papers published in 1869, 1870 (two articles), and 1877
obtained the same results as Riemann when studying the transformation of the
quadratic differential formula gijdx
idxj to the Euclidian metric ds2 ¼
X
i
dxi
� �2
.
The curvature analysis of the Riemann space EN was carried out by Ricci-Curbastro
and Levi-Civita who deducted the expression of the curvature tensor in a very
formal and concise approach, which was also obtained by Christoffel, whose
deduction has an extensive algebrism. In 1917 Tulio Levi-Civita, and after Jan
Arnoldus Schouten (1918) and Karl Hessenberg found independently an interpre-
tation for the curvature tensor associating it to the concept of parallel transport of
vectors.
5.2.1 Formulation
The covariant derivative of a tensor is a tensor, just as when repeating this
differentiation will provide a new tensor. However, the differentiation order with
respect to the variables must be considered in this analysis.
For a function ϕ(xi) of class C2 that represents a scalar field exists the derivative
∂ϕ xið Þ
∂xk that represents a covariant vector. Differentiating again with respect to the
variable xj results by means of the partial differentiation rule of Differential
Calculus
∂2ϕ xið Þ
∂xk∂xj
¼ ∂
2ϕ xið Þ
∂xj∂xk
In this case, the covariant derivative is commutative. However, for tensors which
components are functions of class C2 represented in curvilinear coordinate systems
this independence of the differentiation order in general is not verified. It is
concluded that only the condition of the functions being class C2 is not enough to
ensure this independence.
228 5 Riemann Spaces
For the case of a covariant vector ui the result for its covariant derivative is the
tensor with variance (0, 2):
∂jui ¼ ∂ui∂xj � u‘Γ
‘
ij ð5:2:1Þ
and with
∂jui ¼ Tij
it follows that for the covariant derivative of this tensor with respect to the variable xk
∂kTij ¼ ∂Tij∂xk � T‘jΓ
‘
ik � Ti‘Γ ‘jk ð5:2:2Þ
The substitution of expression (5.2.1) provides
∂kTij ¼
∂ ∂jui
� �
∂xk
� ∂ju‘
� �
Γ ‘ik � ∂‘uið ÞΓ ‘jk
¼ ∂
∂xk
∂ui
∂xj
� u‘Γ ‘ij
� �
� ∂u‘
∂xj
� umΓm‘j
� �
Γ ‘ik �
∂ui
∂x‘
� umΓmi‘
� �
Γ ‘jk
it follows that
∂kTij ¼ ∂j∂kui ¼ ∂
2
ui
∂xk∂xj
� ∂u‘
∂xk
Γ ‘ij � u‘
∂Γ ‘ij
∂xj
� ∂u‘
∂xj
Γ ‘ik
þ umΓm‘jΓ ‘ik �
∂ui
∂x‘
Γ ‘jk þ umΓmi‘Γ ‘jk
ð5:2:3Þ
that represents a tensor with variance (0, 3).
The inversion of the differentiation order provides
∂k∂jui ¼ ∂
2
ui
∂xj∂xk
� ∂u‘
∂xj
Γ ‘ik � u‘
∂Γ ‘ik
∂xj
� ∂u‘
∂xk
Γ ‘ij þ umΓm‘kΓ ‘ij �
∂ui
∂x‘
Γ ‘kj
þ umΓmi‘Γ ‘kj ð5:2:4Þ
In Differential Calculus the differentiation order does not change the result
obtained then
∂2ui
∂xj∂xk
¼ ∂
2
ui
∂xk∂xj
5.2 The Curvature Tensor 229
Subtracting expression (5.2.4) from expression (5.2.3) and considering the
symmetry of the Christoffel symbols
∂j∂kui � ∂k∂jui ¼ u‘ ∂Γ
‘
ik
∂xj
� ∂Γ
‘
ij
∂xk
 !
þ um Γm‘jΓ ‘ik � Γm‘kΓ ‘ij
� �
and with the permutation of the dummy indexes m $ ‘ in the second term to
the right
∂j∂kui � ∂k∂jui ¼ u‘ ∂Γ
‘
ik
∂xj
� ∂Γ
‘
ij
∂xk
 !
þ Γ ‘mjΓmik � Γ ‘mkΓmij
� �" #
Putting
R ‘ijk ¼
∂Γ ‘ik
∂xj
� ∂Γ
‘
ij
∂xk
þ Γ ‘mjΓmik � Γ ‘mkΓmij ð5:2:5Þ
results in
∂j∂kui � ∂kjui ¼ u‘R ‘ijk
The quotient law is used for verifying if the variety R‘ijk is a tensor, carrying out
the inner product of vector u‘ by R
‘
ijk:
R ‘ijku‘ ¼ R ‘ijk‘ ¼ Rijk
The transformation law of tensors to the variety Rijk is given by
Rpqr ¼ ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
Rijk
for the vector u‘ the result of the transformation law is
um ¼ ∂x
‘
∂xm
u‘
that substituted in previous expression provides
Rpqr ¼ ∂x
i
∂xp
∂xj
∂xq
∂xk
∂xr
∂x‘
∂xm
R ‘ijku‘
In the coordinate system X
i
the variety Rpqr is given by
230 5 Riemann Spaces
Rpqr ¼ R ‘pqru‘
whereby
R
‘
pqr �
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
∂x‘
∂xm
R ‘ijk
� �
u‘ ¼ 0
As �u‘ is an arbitrary vector it results in
R
‘
pqr ¼
∂xi
∂xp
∂xj
∂xq
∂xk
∂xr
∂x‘
∂xm
R ‘ijk
that represents the transformation law of tensor with variances (1, 3), as R‘ijk is a
tensor. The tensor defined by expression (5.2.5) is called Riemann–Christoffel
curvature tensor, Riemann–Christoffel mixed tensor, or Riemann–Christoffel ten-
sor of the second kind, or simply curvature tensor. This tensor defines a tensorial
field that depends only on the metric tensor and its derivatives up to the second
order, and classifies the space, for thus R ‘ijk 6¼ 0 the result is a space with curvature.
5.2.2 Differentiation Commutativity
The formulation of an analogous analysis for a contravariant vector ui, which
generates a mixed tensor with variance (1, 1), is carried out by calculating firstly
the covariant derivative of this vector with respect to the coordinate xj:
∂ju
i ¼ ∂u
i
∂xj
þ u‘Γ ij‘ ¼ T ij ð5:2:6Þ
The covariant derivative of the second order of this vector with respect to the
coordinate xk is given by
∂k ∂ju
i
� � ¼ ∂kT ij ¼ ∂T ij∂xk þ T ‘j Γ i‘k � T i‘Γ ‘jk
Substituting expression (5.2.6) in this expression
∂k ∂ju
i
� � ¼ ∂
∂xk
∂ui
∂xj
þ u‘Γ ij‘
� �
þ ∂u
‘
∂xj
þ umΓ ‘jm
� �
Γ i‘k �
∂ui
∂x‘
þ umΓ i‘m
� �
Γ ‘jk
whereby
5.2 The Curvature Tensor 231
∂k ∂ju
i
� � ¼ ∂2ui
∂xk∂xj
þ ∂u
‘
∂xk
Γ ij‘ þ u‘
∂Γ ij‘
∂xk
þ ∂u
‘
∂xj
Γ i‘k þ umΓ ‘jmΓ i‘k �
∂ui
∂x‘
Γ ‘jk
� umΓ i‘mΓ ‘jk ð5:2:7Þ
The inversion of the differentiation is obtained interchanging indexes j $ k, so
∂j ∂ku
i
� � ¼ ∂2ui
∂xj∂xk
þ ∂u
‘
∂xj
Γ ik‘ þ u‘
∂Γ ik‘
∂xj
þ ∂u
‘
∂xk
Γ i‘j þ umΓ ‘kmΓ i‘j �
∂ui
∂x‘
Γ ‘kj
� umΓ i‘mΓ ‘kj ð5:2:8Þ
As in the partial derivative the order of differentiation does not change the result
∂2ui
∂xk∂xj
¼ ∂
2
ui
∂xj∂xk
and subtracting expression (5.2.7) from expression (5.2.8)
∂k ∂ju
i
� �� ∂j ∂kui� � ¼ ∂2ui∂xk∂xj þ ∂u‘∂xk Γ ij‘ þ u‘ ∂Γ
i
j‘
∂xk
þ ∂u
‘
∂xj
Γ i‘k
 
þ umΓ ‘jmΓ i‘k �
∂ui
∂x‘
Γ ‘jk �umΓ i‘mΓ ‘jk
�
� ∂
2
ui
∂xj∂xk
þ ∂u
‘
∂xj
Γ ik‘ þ u‘
∂Γ ik‘
∂xj
þ ∂u
‘
∂xk
Γ i‘j
 
þ umΓ ‘kmΓ i‘j �
∂ui
∂x‘
Γ ‘kj � umΓ i‘mΓ ‘kj
�
and with the symmetry of the Christoffel symbols
∂k ∂ju
i
� �� ∂j ∂kui� � ¼ u‘ ∂Γ ij‘∂xk � ∂Γ ik‘∂xj
 !
þ um Γ ‘jmΓ i‘k � Γ ‘kmΓ i‘j
� �
The permutation of indexes ‘ $ m in the last two terms provides
∂k ∂ju
i
� �� ∂j ∂kui� � ¼ ∂Γ ij‘∂xk � ∂Γ ik‘∂xj þmj‘ Γ imk � Γmk‘Γ imj
 !
u‘
232 5 Riemann Spaces
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and putting
Ri‘kj ¼
∂Γ ij‘
∂xk
� ∂Γ
i
k‘
∂xj
þ Γmj‘Γ imk � Γmk‘Γ imj ð5:2:9Þ
it results in
∂k ∂ju
i
� �� ∂j ∂kui� � ¼ Ri‘kju‘ ð5:2:10Þ
The permutation of indexes j $ k provides
∂j ∂ku
i
� �� ∂k ∂jui� � ¼ Ri‘jku‘
where
Ri‘jk ¼
∂Γ ik‘
∂xj
� ∂Γ
i
j‘
∂xk
þ Γmk‘Γ imj � Γmj‘Γ imk ð5:2:11Þ
This analysis shows that Ri‘jk ¼ 0 ) ∂j∂kuk ¼ ∂k∂juk, i.e., the space is flat. The
necessary and sufficient condition so that the differentiation commutativity be valid
is that the tensor Ri‘jk be null.
5.2.3 Antisymmetry of Tensor Ri‘jk
The comparison of expressions (5.2.9) and (5.2.11) shows that the Riemann–
Christoffel curvature tensor is antisymmetric with respect to the last two indexes
Ri‘kj ¼ �Ri‘jk
5.2.4 Notations for Tensor Ri‘jk
Putting the indexes in the sequence i, j, k, ‘ the result in tensorial notation is
R ¼ R ‘ijkg‘ � gk � gj � gi ð5:2:12Þ
and rewriting the Riemann–Christoffel curvature tensor as
5.2 The Curvature Tensor 233
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R ‘ijk ¼
∂Γ ‘ik
∂xj
� ∂Γ
‘
ij
∂xk
þ ΓmikΓ ‘mj � Γmij Γ ‘mk
the result in symbolic form by means of determinants is
R ‘ijk ¼
∂
∂xj
∂
∂xk
Γ ‘ij Γ
‘
ik
						
						þ
Γmik Γ
m
ij
Γ ‘mk Γ
‘
mj
					
					 ð5:2:13Þ
5.2.5 Uniqueness of Tensor R‘ijk
The metric tensor gij and its conjugated tensor g
ij are unique in a Riemann space,
then their partial derivatives of the first and second order the Christoffel symbols of
this space are unique at pointxi2EN . Thus it is verified that expression (5.2.11) does
not ensure that tensor Ri‘jk is the only tensor that can be expressed by the derivatives
of the first and second order of the metric tensor.
However, the covariant derivatives of a contravariant vector with respect to the
coordinates of a referential system are unique at point xi2EN , and having the
Riemann–Christoffel curvature tensor with variance (1, 3) obtained by means of
these derivatives, it is concluded that it is unique in the point being considered.
Expressions (5.2.5) and (5.2.11) obtained in distinct manners indicate this tensor’s
uniqueness.
For the points xi2EN in which the Christoffel symbols are null, it is verified that
Ri‘jk is expressed by means of a linear combination of the derivatives of the second
order of the metric tensor.
5.2.6 First Bianchi Identity
The Riemann–Christoffel curvature tensor
R ‘ijk ¼
∂Γ ‘ik
∂xj
� ∂Γ
‘
ij
∂xk
þ ΓmikΓΓ ‘mj � Γmij Γ ‘mk
and the cyclic permutations of indexes i, j, k generate the expressions
R ‘jki ¼
∂Γ ‘ji
∂xk
� ∂Γ
‘
jk
∂xi
þ Γmji Γ ‘mk � ΓmjkΓ ‘mi
234 5 Riemann Spaces
R ‘kij ¼
∂Γ ‘kj
∂xi
� ∂Γ
‘
ki
∂xj
þ ΓmkjΓ ‘mi � ΓmkiΓ ‘mj
The sum of these three expressions provides the first Bianchi identity for the
Riemann–Christoffel curvature tensor
R ‘ijk þ R ‘jki þ R ‘kij ¼ 0 ð5:2:14Þ
5.2.7 Second Bianchi Identity
The covariant derivative of a tensor with variance (1, 3) is given by
∂kT
j
p‘m ¼
∂T jp‘m
∂xk
� T jq‘mΓ qpk � T jpqmΓ q‘k � T jp‘qΓ qmk þ T qp‘mΓ jkq
whereby for the Riemann–Christoffel curvature tensor yields it follows that
R ‘ijk ¼
∂Γ ‘ik
∂xj
� ∂Γ
‘
ij
∂xk
þ ΓmikΓ ‘mj � Γmij Γ ‘mk
The covariant derivative with respect to the coordinate xp is given by
∂pR ‘ijk ¼
∂2Γ ‘ik
∂xp∂xj
� ∂
2Γ ‘ij
∂xp∂xk
þ ∂Γ
m
ik
∂xp
Γ ‘mj þ Γmik
∂Γ ‘mj
∂xp
� ∂Γ
m
ij
∂xp
Γ ‘mk � Γmij
∂Γ ‘mk
∂xp
þRmijkΓ ‘mp � R ‘mjkΓmip � R ‘imkΓmjp � R ‘ijmΓmkp
and with the cyclic permutation of indexes j, k, p it follows that
∂jR ‘ikp ¼
∂2Γ ‘ip
∂xj∂xk
� ∂
2Γ ‘ik
∂xj∂xp
þ ∂Γ
m
ip
∂xj
Γ ‘mk þ Γmip
∂Γ ‘mk
∂xj
� ∂Γ
m
ik
∂xj
Γ ‘mp � Γmik
∂Γ ‘mp
∂xj
þ RmikpΓ ‘mj � R ‘mkpΓΓmij � R ‘impΓmkj � R ‘ikmΓmpj
∂kR ‘ipj ¼
∂2Γ ‘ij
∂xk∂xp
� ∂
2Γ ‘ip
∂xk∂xj
þ ∂Γ
m
ij
∂xk
Γ ‘mp þ Γmij
∂Γ ‘mp
∂xk
� ∂Γ
m
ip
∂xk
Γ ‘mj � Γmip
∂Γ ‘mj
∂xk
þ RmipjΓ ‘mk � R ‘mpjΓmik � R ‘imjΓmpk � R ‘ipmΓmjk
The sum of these three expressions provides
5.2 The Curvature Tensor 235
∂pR ‘ikj þ ∂jR ‘ikp þ ∂kR ‘ipj ¼
∂2Γ ‘ik
∂xp∂xj
� ∂
2Γ ‘ij
∂xp∂xk
þ ∂Γ
m
ik
∂xp
Γ ‘mj þ Γmik
∂Γ ‘mj
∂xp
� ∂Γ
m
ij
∂xp
Γ ‘mk
�Γmij
∂Γ ‘mk
∂xp
þ RmijkΓ ‘mp � R ‘mjkΓmip � R ‘imkΓmjp � R ‘ijmΓmkp
þ ∂
2Γ ‘ip
∂xj∂xk
� ∂
2Γ ‘ik
∂xj∂xp
þ ∂Γ
m
ip
∂xj
Γ ‘mk þ Γmip
∂Γ ‘mk
∂xj
� ∂Γ
m
ik
∂xj
Γ ‘mp
�Γmik
∂Γ ‘mp
∂xj
þ RmikpΓ ‘mj � R ‘mkpΓmij � R ‘impΓmkj � R ‘ikmΓmpj
þ ∂
2Γ ‘ij
∂xk∂xp
� ∂
2Γ ‘ip
∂xk∂xj
þ ∂Γ
m
ij
∂xk
Γ ‘mp þ Γmij
∂Γ ‘mp
∂xk
� ∂Γ
m
ip
∂xk
Γ ‘mj
�Γmip
∂Γ ‘mj
∂xk
þ RmipjΓ ‘mk � R ‘mpjΓmik � R ‘imjΓmpk � R ‘ipmΓmjk
and with the equalities
∂2Γ ‘ik
∂xp∂xj
¼ ∂
2Γ ‘ik
∂xj∂xp
∂2Γ ‘ij
∂xp∂xk
¼ ∂
2Γ ‘ij
∂xk∂xp
∂2Γ ‘ip
∂xj∂xk
¼ ∂
2Γ ‘ip
∂xk∂xj
the previous expression stays
∂pR ‘ikj þ ∂jR ‘ikp þ ∂kR ‘ipj ¼
∂Γmik
∂xp
Γ ‘mj þ Γmik
∂Γ ‘mj
∂xp
� ∂Γ
m
ij
∂xp
Γ ‘mk � Γmij
∂Γ ‘mk
∂xp
þRmijkΓ ‘mp � R ‘mjkΓmip � R ‘imkΓmjp � R ‘ijmΓmkp
þ∂Γ
m
ip
∂xj
Γ ‘mk þ Γmip
∂Γ ‘mk
∂xj
� ∂Γ
m
ik
∂xj
Γ ‘mp � Γmik
∂Γ ‘mp
∂xj
þRmikpΓ ‘mj � R ‘mkpΓmij � R ‘impΓmkj � R ‘ikmΓmpj
þ∂Γ
m
ij
∂xk
Γ ‘mp þ Γmij
∂Γ ‘mp
∂xk
� ∂Γ
m
ip
∂xk
Γ ‘mj � Γmip
∂Γ ‘mj
∂xk
þRmipjΓ ‘mk � R ‘mpjΓmik � R ‘imjΓmpk � R ‘ipmΓmjk
Putting the Christoffel symbols in evidence and considering the antisymmetry of
the Riemann–Christoffel curvature tensor, i.e., R ‘imk ¼ �R ‘ikm, R ‘ijm ¼ �R ‘imj,
R ‘imp ¼ �R ‘ipm, and the symmetry of the Christoffel symbols, i.e., Γmjp ¼ Γmpj , Γmkp
¼ Γmpk, Γmkj ¼ Γmjk it follows that
236 5 Riemann Spaces
∂pR ‘ikjþ∂jR ‘ikpþ∂kR ‘ipj ¼ Γ ‘mp Rmijk�
∂Γmik
∂xj
þ∂Γ
m
ij
∂xk
� �
þΓ ‘mj Rmikpþ
∂Γmik
∂xp
�∂Γ
m
ip
∂xk
� �
þΓ ‘mk Rmipj�
∂Γmij
∂xp
þ∂Γ
m
ip
∂xj
� �
�Γmip R ‘mjk�
∂Γ ‘mk
∂xj
þ∂Γ
‘
mj
∂xk
 !
�Γmij R ‘mkpþ
∂Γ ‘mk
∂xp
�∂Γ
‘
mp
∂xk
 !
�Γmik R ‘mpj�
∂Γ ‘mj
∂xp
þ∂Γ
‘
mp
∂xj
 !
The expressions of the tensors are given by
Rmijk ¼
∂Γmik
∂xj
� ∂Γ
m
ij
∂xk
þ Γ qikΓmqj � Γ qijΓmqk
Rmikp ¼
∂Γmip
∂xk
� ∂Γ
m
ik
∂xp
þ Γ qipΓmqk � Γ qikΓmqp
Rmipj ¼
∂Γmij
∂xp
� ∂Γ
m
ip
∂xj
þ Γ qijΓmqp � Γ qipΓmqj
R ‘mjk ¼
∂Γ ‘mk
∂xj
� ∂Γ
‘
mj
∂xk
þ Γ qmkΓ ‘qj � Γ qmjΓ ‘qk
R ‘mkp ¼
∂Γ ‘mp
∂xk
� ∂Γ
‘
mk
∂xp
þ Γ qmpΓ ‘qk � Γ qmkΓ ‘qp
R ‘mpj ¼
∂Γ ‘mj
∂xp
� ∂Γ
‘
mp
∂xj
þ Γ qmjΓ ‘qp � Γ qmpΓ ‘qj
that substituted in previous expression provide
∂pR ‘ikjþ∂jR ‘ikpþ∂kR ‘ipj ¼ Γ ‘mp
∂Γmik
∂xj
�∂Γ
m
ij
∂xk
þΓ qikΓmqj �Γ qijΓmqk�
∂Γmik
∂xj
þ∂Γ
m
ij
∂xk
� �
þΓ ‘mj
∂Γmip
∂xk
�∂Γ
m
ik
∂xp
þΓ qipΓmqk�Γ qikΓmqpþ
∂Γmik
∂xp
�∂Γ
m
ip
∂xk
� �
þΓ ‘mk
∂Γmij
∂xp
�∂Γ
m
ip
∂xj
þΓ qijΓmqp�Γ qipΓmqj �
∂Γmij
∂xp
þ∂Γ
m
ip
∂xj
� �
�Γmip
∂Γ ‘mk
∂xj
�∂Γ
‘
mj
∂xk
þΓ qmkΓ ‘qj�Γ qmjΓ ‘qk�
∂Γ ‘mk
∂xj
þ∂Γ
‘
mj
∂xk
 !
�Γmij
∂Γ ‘mp
∂xk
�∂Γ
‘
mk
∂xp
þΓ qmpΓ ‘qk�Γ qmkΓ ‘qpþ
∂Γ ‘mk
∂xp
�∂Γ
‘
mp
∂xk
 !
�Γmik
∂Γ ‘mj
∂xp
�∂Γ
‘
mp
∂xj
þΓ qmjΓ ‘qp�Γ qmpΓ ‘qj�
∂Γ ‘mj
∂xp
þ∂Γ
‘
mp
∂xj
 !
5.2 The Curvature Tensor 237
Simplifying
∂pR ‘ikj þ ∂jR ‘ikp þ ∂kR ‘ipj ¼ Γ ‘mp Γ qikΓmqj � Γ qijΓmqk
� �
þΓ ‘mj Γ qipΓmqk � Γ qikΓmqp
� �
þΓ ‘mk Γ qijΓmqp � Γ qipΓmqj
� �
�Γmip Γ qmkΓ ‘qj � Γ qmjΓ ‘qk
� �
�Γmij Γ qmpΓ ‘qk � Γ qmkΓ ‘qp
� �
�Γmik Γ qmjΓ ‘qp � Γ qmpΓ ‘qj
� �
and with the permutation of the dummy indexes m $ q in the first six terms it
follows that
∂pR ‘ikj þ ∂jR ‘ikp þ ∂kR ‘ipj ¼ Γ ‘qpΓmikΓ qmj � Γ ‘qpΓmij Γ qmk
þΓ ‘qjΓmipΓ qmk � Γ ‘qjΓmikΓ qmp
þΓ ‘qkΓmij Γ qmp � Γ ‘qkΓmipΓ qmj
�ΓmipΓ qmkΓ ‘qj þ ΓmipΓ qmjΓ ‘qk
�Γmij Γ qmpΓ ‘qk þ Γmij Γ qmkΓ ‘qp
�ΓmikΓ qmjΓ ‘qp þ ΓmikΓ qmpΓ ‘qj
whereby
∂pR
‘
ikj þ ∂jR ‘ikp þ ∂kR ‘ipj ¼ 0 ð5:2:15Þ
that is called second Bianchi identity.
5.2.8 Curvature Tensor of Variance (0, 4)
The Riemann–Christoffel curvature tensor generates a curvature tensor expressed
in covariant components. With the multiplying of tensor R‘ijk by the metric tensorgp‘ it follows that
238 5 Riemann Spaces
gp‘R
‘
ijk ¼ gp‘
∂Γ ‘ik
∂xj
� ∂Γ
‘
ij
∂xk
þ ΓmikΓ ‘mj � Γmij Γ ‘mk
 !
or
gp‘R
‘
ijk ¼
∂ gp‘Γ
‘
ik
� �
∂xj
� ∂gp‘
∂xj
Γ ‘ik �
∂ gp‘Γ
‘
ij
� �
∂xk
þ ∂gp‘
∂xk
Γ ‘ij þ gp‘ΓmikΓΓ ‘mj � gp‘Γmij Γ ‘mk
Ricci’s identity allows writing
∂gp‘
∂xj
¼ Γpj, ‘ þ Γ‘j, p
∂gp‘
∂xk
¼ Γpk, ‘ þ Γ‘k, p
then
gp‘R
‘
ijk ¼
∂ gp‘Γ
‘
ik
� �
∂xj
�
∂ gp‘Γ
‘
ij
� �
∂xk
þ Γ ‘ik Γpj, ‘ þ Γ‘j, p
� �þ Γ ‘ij Γpk, ‘ þ Γ‘k, p� �
�Γmk, pΓmij þ Γmj, pΓmik
¼ ∂Γik, p
∂xj
� ∂Γij, p
∂xk
� Γ ‘ik Γpj, ‘ þ Γ‘j, p
� �þ Γ ‘ij Γpk, ‘ þ Γ‘k, p� �
�Γmk, pΓmij þ Γmj, pΓmik
and replacing indexes m ! ‘ in the last two terms
gp‘R
‘
ijk ¼
∂Γik,p
∂xj
�∂Γij,p
∂xk
�Γ ‘ik Γpj,‘þΓ‘j,p
� �þΓ ‘ij Γpk,‘þΓ‘k,p� ��Γ‘k,pΓ ‘ijþΓ‘j,pΓ ‘ik
¼∂Γik,p
∂xj
�∂Γij,p
∂xk
þΓ ‘ijΓpk,‘�Γ ‘ikΓpj,‘
whereby the result for the Riemann–Christoffel curvature tensor with variance
(0, 4) or Riemann–Christoffel of first tensor type is
Rpijk ¼ ∂Γik, p∂xj �
∂Γij, p
∂xk
þ Γ ‘ijΓpk, ‘ � Γ ‘ikΓpj, ‘ ð5:2:16Þ
which in tensorial notation is written as
R ¼ Rpijkgp � gi � gj � gk ð5:2:17Þ
and in symbolic form by means of determinants stays
5.2 The Curvature Tensor 239
Rpijk ¼
∂
∂xj
∂
∂xk
Γij, p Γik, p
						
						þ Γ
‘
ij Γ
‘
ik
Γpj, ‘ Γpk, ‘
					
					 ð5:2:18Þ
In tensorial notation the Riemann–Christoffel tensors, mixed and covariant, are
represented by R.
5.2.9 Properties of Tensor Rpijk
For the Riemann–Christoffel covariant tensor the first Bianchi identity provides
g‘p R
‘
ikj þ R ‘jki þ R ‘kij
� �
¼ 0
whereby the following cyclic property results
Rpikj þ Rpjki þ Rpkij ¼ 0 ð5:2:19Þ
Considering the antisymmetry of the Riemann–Christoffel tensor with variance
(1, 3) the result is
gp‘R
‘
ijk ¼ �gp‘R ‘ikj ) Rpijk ¼ �Rpikj
then the Riemann–Christoffel tensor with variance (0, 4) is antisymmetric in the last
two indexes.
Rewriting expression (5.2.16)
Rpijk ¼ ∂Γik, p∂xj �
∂Γij, p
∂xk
þ Γ ‘ijΓpk, ‘ � Γ ‘ikΓpj, ‘
and with expressions
Γik, p ¼ 1
2
∂gpk
∂xi
þ ∂gip
∂xk
� ∂gik
∂xp
� �
Γij, p ¼ 1
2
∂gjp
∂xi
þ ∂gip
∂xj
� ∂gij
∂xp
� �
Γpk, ‘ ¼ gq‘Γ qpk Γpj, ‘ ¼ gq‘Γ qpj
it follows that
Rpijk ¼ ∂∂xj
1
2
∂gpk
∂xi
þ ∂gip
∂xk
� ∂gik
∂xp
� �
 �
� ∂
∂xk
1
2
∂gjp
∂xi
þ ∂gip
∂xj
� ∂gij
∂xp
� �
 �
þ gq‘Γ qpkΓ ‘ij � gq‘Γ qpjΓ ‘ik
240 5 Riemann Spaces
Rpijk ¼ 1
2
∂2gik
∂xj∂xp
þ ∂
2
gpk
∂xj∂xi
� ∂
2
gji
∂xk∂xp
� ∂
2
gpj
∂xk∂xi
 !
þ gq‘ Γ qpkΓ ‘ij � Γ qpjΓ ‘ik
� �
ð5:2:20Þ
The expression (5.2.20) allows calculating the components of the tensor Rpijk
directly in terms of the metric tensor. With the permutation of indexes i $ p in
expression (5.2.20)
Ripjk ¼ 1
2
∂2gpk
∂xj∂xi
þ ∂
2
gik
∂xj∂xp
� ∂
2
gjp
∂xk∂xi
� ∂
2
gij
∂xk∂xp
 !
þ gq‘ Γ qpkΓ ‘ij � Γ qpjΓ ‘ik
� �
and with the permutation of the dummy indexes q $ ‘ this expression becomes
Ripjk ¼ 1
2
∂2gpk
∂xj∂xi
þ ∂
2
gik
∂xj∂xp
� ∂
2
gjp
∂xk∂xi
� ∂
2
gij
∂xk∂xp
 !
þ g‘q Γ ‘ikΓ qpj � Γ ‘ijΓ qpk
� �
Considering the symmetry of the metric tensor it is verified that the term to the
right represents the components �Rpijk, then Ripjk ¼ �Rpijk, i.e., the tensor is
antisymmetric in the first two indexes. These analyses show that the tensor Rpijk
is antisymmetric in the first two and the last two indexes.
The permutation of indexes p $ j, i $ k in expression (5.2.20) leads to
Rpijk ¼ 1
2
∂2gji
∂xp∂xk
� ∂
2
gki
∂xp∂xj
� ∂
2
gjp
∂xi∂xk
þ ∂
2
gpk
∂xi∂xj
 !
þ gq‘ Γ qjiΓ ‘kp � Γ qjpΓ ‘ki
� �
The symmetry of the metric tensor gives Rpijk ¼ Rjkpi. It is concluded that the
tensor Rpijk is symmetric for the permutation of the pair of initial indexes for the pair
of final indexes.
5.2.10 Distinct Algebraic Components of Tensor Rpijk
The number of components of tensor Rpijk in the Riemann space EN cannot be
obtained counting the equations Rpikj þ Rpjki þ Rpkij ¼ 0 and considering the com-
ponents antisymmetricRpijk ¼ �Ripjk,Rpijk ¼ �Rpikj and the symmetric components
Rpijk ¼ Rjkpi, because these two equations overlap. The methodology used to carry
out this counting is given by means of classifying the tensor components into four
groups, as a function of the number of repeated indexes:
(a) The four indexes are equal Riiii
(b) The initial pair of indexes is equal to the second pair Ripip
5.2 The Curvature Tensor 241
(c) One index is repeated Rppik
(d) The four indexes are different Rpijk
Case (a) must fulfill the antisymmetry of tensor Rpijk that provides Riiii ¼ �Riiii
then Riiii ¼ 0. The components are null when the four indexes are equal.
For case (b) only two indexes are different: Ripip having that these components
differ from the components Rippi solely in the sign, and by the antisymmetry the
result is Rpipi ¼ �Rippi ¼ � �Ripip
� � ¼ Ripip. There is a number of components for
Ripip as many as the different pair of indexes, i.e., i 6¼ p. For index i there are
N distinct combinations, and for index p there are N � 1ð Þ distinct combinations,
and considering the antisymmetry of the tensor for these last indexes N
2
N � 1ð Þ
different combinations result. This number of combinations corresponds to the
number of the N
2
N � 1ð Þ distinct combinations. There is no reduction of com-
ponents due to the symmetry Rpijk ¼ Rjkpi. The first Bianchi identity is satisfied, for
Rpipi þ Rppii þ Rpiip ¼ Rpipi þ 0� Rpipi ¼ 0
does not reduce the number of components. Therefore, in this case only N
2
N � 1ð Þ
independent components are non-null.
Case (c) has components of the kind Rppik. In this case there are N combinations
for the index p, N � 1ð Þ combinations for index i, and N � 2ð Þ combinations for
index k. The number of combinations for the indexes provides the number of tensor
components. The antisymmetry does not reduce the number of components, for
Rppik ¼ 0 and Rpipk ¼ 0, and the first Bianchi identity is satisfied. Considering the
symmetryRpipk ¼ Rpkpi the number of components is reduced by half, whereby there
are N
2
N � 1ð Þ N � 2ð Þ independent and non-null components. Admitting the four
indexes different there are, for example, the components R1234,R2314,R3124.
With methodology analogous to the previous case, it is verified that the indexes
p, i, j, k can be selected in N N � 1ð Þ N � 2ð Þ N � 3ð Þ modes. Considering the
antisymmetries Rpijk ¼ �Ripjk and Rpijk ¼ �Rpikj, the combination of indexes is
reduced to N
4
N � 1ð Þ N � 2ð Þ N � 3ð Þ modes. The symmetry Rpijk ¼ Rjkpi reduces to
half these combinations, then having N
8
N � 1ð Þ N � 2ð Þ N � 3ð Þ modes. The first
Bianchi identity is given by
Rpikj þ Rpjki þ Rpkij ¼ 0 ) Rpikj ¼ � Rpjki þ Rpkij
� �
that shows that the different combinations of the indexes are related among
themselves, for a component can be expressed in terms of the other two. Therefore,
the total number of combinations of indexes is reduced in 2
3
, and the total number of
non-null independent components for this case is 2
3
N
8
N � 1ð Þ N � 2ð Þ N � 3ð Þ.
The consideration of all the cases that were analyzed leads to
242 5 Riemann Spaces
0þ N
2
N � 1ð Þ þ N
2
N � 1ð Þ N � 2ð Þ þ N
12
N � 1ð Þ N � 2ð Þ N � 3ð Þ
whereby there are N
2
12
N2 � 1� � independent and non-null components for the tensor
Rpijk.
The expressions that provide the Christoffel symbols for the orthogonal coordi-
nate systems are
Γ kij ¼ 0 Γ kii ¼ �
1
2gkk
∂gii
∂xk
Γ iij ¼
∂ ‘n
ffiffiffiffiffi
gii
p� �
∂xj
Γ iii ¼
∂ ‘n
ffiffiffiffiffi
gii
p� �
∂xi
and with expression (5.2.20) that defines the Riemann–Christoffel curvature tensor
with variance (0, 4) it results for the components of this tensor, where the indexes
p, i, j, k indicate no summation:
– Four different indexes
Rpijk ¼ 0 ð5:2:21Þ
– i ¼ j and the other three indexes different
Rpiik ¼ ffiffiffiffiffigiip ∂2 ffiffiffiffiffigiip∂xp∂xk � ∂
ffiffiffiffiffi
gii
p
∂xp
∂ ‘n ffiffiffiffiffiffigppp� �
∂xk
� ∂
ffiffiffiffiffi
gii
p
∂xk
∂ ‘n
ffiffiffiffiffiffi
gkk
p� �
∂xp
0@ 1A ð5:2:22Þ
– p ¼ k, i ¼ j, p 6¼ i (two different indexes)
Rkiik ¼ ffiffiffiffiffigiip ffiffiffiffiffiffigkkp ∂∂xk 1ffiffiffiffiffiffigkkp∂
ffiffiffiffiffi
gii
p
∂xk
� �
þ ∂
∂xi
1ffiffiffiffiffi
gii
p ∂
ffiffiffiffiffiffi
gkk
p
∂xi
� �
þ 1
gmm
∂
ffiffiffiffiffi
gii
p
∂xm
∂
ffiffiffiffiffiffi
gkk
p
∂xm
 �
ð5:2:23Þ
withm 6¼ pandm ¼ i to fulfill the condition of having two different pairs of indexes,
with the summation carried out only for the index m.
Table 5.1 shows four Riemann spaces EN and the independent and non-null
components of tensor Rpijk.
For the Riemann space E1 the only component of tensor Rpijk is R1111, which by
means of its antisymmetry will always be null. Expression N
2
12
N2 � 1� � proves this
nullity. It is concluded that this tensor express only the internal properties of the
space and not the way how this space is embedded in the Riemann spaces EN,
N > 1, for this characteristic verifies that in E1 a curved line has null curvature, seen
that R1111 ¼ 0.
5.2 The Curvature Tensor 243
For the Riemann space E2 the tensor Rpijk has null components when three or
more indexes are equal.
Only one component cannot be null: R1212. By means of the symmetry and the
antisymmetry it is verified that R1212 ¼ �R2112 ¼ �R1221 ¼ R2121. This component
is given by
R1212 ¼ 1
2
2
∂2g12
∂x1∂x2
� ∂
2
g11
∂x2∂x2
� ∂
2
g22
∂x1∂x1
 !
þ gq‘ Γ q12Γ ‘12 � Γ q11Γ ‘22
� � ð5:2:24Þ
For the Riemann space E3 the six components of tensor Rpijk are:
– Three components with two repeated indexes
R1212 R1313
R2323
– Three components with only one index repeated (three indexes are different)
R1213 R1223(¼R2123) R1323(¼R3132)
For the Riemann space E4 there are 21 non-null components of tensor R‘ijkwhich
are:
– Six components with two repeated indexes
R1212 R1313 R1414
R2323 R2424 R3434
– Twelve components with only one index repeated (three indexes are different)
R1213 R1214 R1223 R1224 R1314 R1323 R1334 R1424 R1434
R2324 R2334 R2434
Table 5.1 Independent and non-null components of tensor Rpijk
Dimension of space EN 2 3 4 5
Number of components 16 81 256 625
Independent and non-null components
of Rpijk
1 6 20 50
Kinds of components R1212 Riþ1 iþ2 jþ1 jþ2 Rpipi,Rppik,
Rpijk
Rpipi,Rppik,
Rpijk
244 5 Riemann Spaces
– Three components with only one index repeated (three indexes are different)
R1234 R1324 R1423
having that R1234 þ R1423 � R1324 ¼ 0, then there are 20 independent non-null
components.
The non-null components of tensor R‘ijk for the Riemann space E5 are:
– Ten components with two repeated indexes
R1212 R1313 R1414 R1515
R2323 R2424 R2525
R3434 R3535
R4545
– Thirty components with only one index repeated (three indexes are different)
R1213 R1214 R1215 R1314 R1315 R1415
R2123 R2124 R2125 R2324 R2325 R2425
R3132 R3134 R3135 R3234 R3235 R3435
R4142 R4143 R4145 R4243 R4245 R4345
R5152 R5153 R5154 R5253 R5254 R5354
– Ten components in which all the indexes are different
R1234 R1235 R1245 R1345 R2345
R1324 R1325 R1425 R1435 R2435
5.2.11 Classification of Spaces
As a function of the values assumed by the Riemann–Christoffel tensors the spaces
are classified as: (a) flat: R ‘ijk ¼ Rijkm ¼ 0; (b) curved space R ‘ijk 6¼ 0; Rijkm 6¼ 0.
The condition Ri‘jk ¼ Rijkm ¼ 0 indicates that the space is flat with the compo-
nents of its metric tensor gij being constant. If the metric ds
2 ¼ gijdxidxj is definite
positive, i.e., gij
			 			 > 0, this space is Euclidian, then it is possible to carry out a linear
transformation of the coordinates xi to the coordinates xi for which the result is
gij ¼ δ ij , so the metric is
ds2 ¼ δ ij dxidxj ¼ dx1dx1 þ dx2dx2 þ � � � þ dxmdxm ð5:2:25Þ
5.2 The Curvature Tensor 245
The vectors of base ei of this new coordinate system X
i form a set of orthogonal
directions, thus δ ij ¼ ei � ei, and define an Euclidian space EM. Consider the Rie-
mann space EN with the coordinates x
i, i ¼ 1, 2, . . .N, EN � EM, with M > N,
which coordinates are xk, k ¼ 1, 2, . . . ,M. Let the functions M be independent in
terms of the coordinates xk, so as to have the metric
ds2 ¼ gijdxidxj ¼ dxk
� � 2 ) gijdxidxj ¼ dxkdxk
By means of the transformation law for coordinates it follows that
dxk ¼ ∂x
k
∂xi
dxi dxk ¼ ∂x
k
∂xj
dxj
gijdx
idxj ¼ ∂x
k
∂xi
dxi
� �
∂xk
∂xj
dxj
� �
) gij �
∂xk
∂xi
∂xk
∂xj
� �
dxidxj ¼ 0
As dxi and dxj are arbitrary, provides
gij ¼
∂xk
∂xi
∂xk
∂xj
that defines N
2
N þ 1ð Þ independent differential equations as a function of
M unknowns xk. In this case M < N
2
N þ 1ð Þ is the condition in order to have
EN � EM. For N ¼ 1 the result is M � N2.
5.3 Riemann Curvature
5.3.1 Definition
The study of the Riemann space EN is carried by means of the definition of the
Riemann K curvature, which is more effective for the formulations of analyses than
the Riemann–Christoffel curvature tensor Rpijk, for it considers the directions of the
space.
For establishing a general formulation, valid for the Riemann spaces EN with
undefined metric, with the unit vectors ui and vi, linearly independents, defined in a
point xi2EN , and the expression
wi ¼ aui þ bvi ð5:3:1Þ
that defines a coplanar vector with these two unit vectors, where a, b, are scalars that
assume arbitrary values. The elementary displacements in the directions defined by
the vectors wi determine a plane π that contains the point xi2EN .
246 5 Riemann Spaces
It is admitted that ui and vi define coplanar vectors
wi ¼ a1ui þ b1vi ð5:3:2Þ
ri ¼ a2ui þ b2vi ð5:3:3Þ
where a1, b1, a2, b2 are scalars, and putting ε uð Þ ¼ 
1 and ε vð Þ ¼ 
1 as functional
indicators of these unit vectors, and having wi and ri vectors mutually orthogonal it
follows that
ε wð Þ ¼ gk‘wkw‘ ¼ a21uku‘ þ b21vkv‘ ¼ a21ε uð Þ þ b21ε vð Þ
ε rð Þ ¼ gk‘rkr‘ ¼ a22ε uð Þ þ b22ε vð Þ
and with the condition of orthogonality
gk‘w
kr‘ ¼ ε uð Þa1a2 þ ε vð Þb1b2 ¼ 0
whereby
ε wð Þε rð Þ ¼ a21ε uð Þ þ b21ε vð Þ
� �
a22ε uð Þ þ b22ε vð Þ
� �� ε uð Þa1a2 þ ε vð Þb1b2½ �2
¼ ε uð Þε vð Þ a1b2 � a2b1ð Þ2
ð5:3:4Þ
As the functional indicators assume the values 
1:
a1b2 � a2b1 ¼ 
1 ð5:3:5Þ
whereby
ε uð Þε vð Þ ¼ ε wð Þε rð Þ ð5:3:6Þ
Consider two orthogonal unit vectors u and v that determine the plane π that
contains the point xi2EN , thus the Riemann curvature is defined by
K ¼ ε uð Þε vð ÞRk‘mnukv‘umvn ð5:3:7Þ
5.3.2 Invariance
For the other pair of orthogonal vectors w and r coplanar with u and v, there is in an
analogous way for the Riemann curvature
eK ¼ ε wð Þε rð ÞRk‘mnwkr‘wmrn
5.3 Riemann Curvature 247
and with expressions (5.3.4)–(5.3.6) it follows that
ε uð Þε vð Þ a1b2 � a2b1ð ÞRk‘mnwkr‘wmrn ¼ ε uð Þε vð ÞRk‘mnukv‘umveK ¼ K
thus the Riemann curvature does not depend on the pair of unit vectors used to
define it, then K is an invariant.
5.3.3 Normalized Form
The obtaining of an expression for the Riemann curvature can be carried out
admitting that the Riemann space EN is isotropic, in which the isotropic tensor is
defined by
Tij‘m ¼ Agijg‘m þ Bgi‘gjm þ Cgimgj‘
where A,B,C are scalars that depend on the point xi2EN .
Assuming that tensor Tij‘m is the curvature tensor Rij‘m the result is
Rij‘m ¼ Agijg‘m þ Bgi‘gjm þ Cgimgj‘ ð5:3:8Þ
and the antisymmetry of tensor Rij‘m allows writing Riiii ¼ 0, Riijj ¼ 0,
Rii‘m ¼ Rij‘‘ ¼ 0, and with expression (5.3.8) it follows that
Riiii ¼ Agiigii þ Bgiigii þ Cgiigii ¼ g2ii Aþ Bþ Cð Þ ¼ 0
Aþ Bþ C ¼ 0 ) Bþ C ¼ �A
Riijj ¼ Agiigjj þ Bgijgij þ Cgijgij ¼ Agiigjj þ Bþ Cð Þgijgij
¼ A giigjj � g2ij
� �
¼ 0 ð5:3:9Þ
Rij‘‘ ¼ Agijg‘‘ þ Bgi‘gj‘ þ Cgi‘gj‘ ¼ A gijg‘‘ � gi‘gj‘
� �
¼ 0 ð5:3:10Þ
The minors of det gi‘ cannot all be simultaneously null, then in expressions
(5.3.9) and (5.3.10) the result is A ¼ 0 and B ¼ �C, whereby
Rij‘k ¼ B gi‘gjm � gimgj‘
� �
ð5:3:11Þ
248 5 Riemann Spaces
Let
K ¼ ε uð Þε vð ÞRk‘mnukv‘umvn
or
K ¼ Rk‘mnukv‘umvn ð5:3:12Þ
and substituting expression (5.3.11) it is concluded that B ¼ K is the Riemann
curvature in xi2EN , so
Rij‘m ¼ K gi‘gjm � gimgj‘
� �
ð5:3:13Þ
The expression of the Riemann curvature for the isotropic space EN, withN > 2,
in terms of the generalized Kronecker delta and the Ricci pseudotensor
εi1i2...imimþ1...inεp1p2...pmpmþ1...pn ¼ N � 2ð Þ!δp1p2...pni1i2...in
takes the form
Rij‘k ¼ K εi1i2...imimþ1...inε
p1p2...pmpmþ1...pn
N � 2ð Þ! ¼ Kδ
p1p2...pn
i1i2...in
ð5:3:14Þ
The normalized Riemann curvature is established admitting that the vectors u
and v form an angle α and define a tangent plane π in point xi2EN . The norm of the
vector perpendicular to this plane is given by
u� vk k2 ¼ uk k2 vk k2 sin 2α
and with the square of the dot product of these two vectors it follows that
u � vð Þ2 ¼ uk k2 vk k2 cos 2α ¼ cos 2α
u� vk k2 ¼ uk k2 vk k2 1� cos 2α� � ¼ uk k2 vk k2 � u � vð Þ2
In terms of the components of these vectors
uk k2 ¼ gkmukum vk k2 ¼ g‘nv‘vn
then
u� vk k2 ¼ gkmukumg‘nv‘vn � gknukvngm‘umv‘ ¼ ukv‘umvn gkmg‘n � gkngm‘ð Þ
Expression (5.3.12) in its normalized form is
5.3 Riemann Curvature 249
K xi; u; v
� � ¼ Rk‘mnukv‘umvn
gkmg‘n � gkngm‘ð Þukv‘umvn
ð5:3:15Þ
or
K xi; u; v
� � ¼ Rk‘mnAk‘Amn
gkmg‘n � gkngm‘ð ÞAk‘Amn
ð5:3:16Þ
where Ak‘ ¼ ukv‘,Amn ¼ umvn represent the plane π defined by the vectors u, v.
This expression highlights that the Riemann curvature K(xi;u, v) of the Riemann
space EN relative to the plane π defined by the vectors u and v depends on the point
xi2π � EN .
In the numerator of expression (5.3.15) the product Rk‘mnu
kv‘umvn is an invariant.
Putting
Gk‘mn ¼ gkmg‘n � gkngm‘
it is verified thatGk‘mn is a tensor, for it is obtained by means of algebraic operations
with the metric tensor. The permutation of the tensor indexes Gk‘mn shows that this
tensor has the same properties of symmetry and antisymmetry as tensor Rk‘mn. For
an orthogonal coordinate system exists gij ¼ 0 for i 6¼ j, and the non-null compo-
nents of this tensor are given by Gijij ¼ giigjj, where the indexes do not indicate
summation.
The inner product Gk‘mnu
kv‘umvn generates a scalar, then expression (5.3.15)
represents an invariant, highlighting the demonstration that eK ¼ K.
5.4 Ricci Tensor and Scalar Curvature
The Riemann–Christoffel curvature tensor Rpijk allows obtaining tensors of lower
order by means of theirs various contractions. To obtain a non-null tensor first an
index of a pair of indexes are contracted with an index of another pair of indexes,
being possible the contractions: 1–3; 1–4; 2–3; 2–4. The contraction of this tensor
generates the Ricci tensor, thus the multiplying of tensor Rpijk by g
mp provides
gmpRpijk ¼ gmpgp‘R ‘ijk ¼ δm‘ R ‘ijk ¼ Rmijk
and with the contraction m ¼ k the result is
Rkijk ¼ Rij
250 5 Riemann Spaces
Then the Riemann–Christoffel curvature tensor with variance (1, 3) provides
two Ricci tensors, one of variance (0, 2) and another of variance (1, 1). The second
contraction gives a scalar with important properties, called scalar curvature. The
Ricci tensor is essentially the only contraction of the Riemann–Christoffel tensor.
5.4.1 Ricci Tensor with Variance (0, 2)
The contraction of the curvature tensor
R ‘ijk ¼
∂Γ ‘ik
∂xj
� ∂Γ
‘
ij
∂xk
þ ΓmikΓ ‘mj � Γmij Γ ‘mk
in indexes ‘ ¼ k provides
Rij ¼ ∂Γ
‘
i‘
∂xj
� ∂Γ
‘
ij
∂x‘
þ Γmi‘Γ ‘mj � Γmij Γ ‘m‘ ð5:4:1Þ
In determinants form the result is
Rij ¼
∂
∂xj
∂
∂x‘
Γ ‘ij Γ
‘
i‘
							
							þ
Γmi‘ Γ
m
ij
Γ ‘m‘ Γ
‘
mj
					
					 ð5:4:2Þ
and with the expressions
Γ ‘i‘ ¼
∂ ‘n
ffiffiffi
g
p� �
∂xi
Γ ‘m‘ ¼
∂ ‘n
ffiffiffi
g
p� �
∂xm
it follows that
Rij ¼ ∂∂xj
∂ ‘n
ffiffiffi
g
p� �
∂xi
 �
� ∂Γ
‘
ij
∂x‘
þ Γmi‘Γ ‘mj � Γmij
∂ ‘n
ffiffiffi
g
p� �
∂xm
whereby for the Ricci tensor with variance (0, 2) the result is
Rij ¼
∂2 ‘n
ffiffiffi
g
p� �
∂xj∂xi
� ∂Γ
‘
ij
∂x‘
þ Γmi‘Γ ‘mj � Γmij
∂ ‘n
ffiffiffi
g
p� �
∂xm
or
5.4 Ricci Tensor and Scalar Curvature 251
Rij ¼ 1
2
∂2 ‘ngð Þ
∂xj∂xi
� ∂Γ
‘
ij
∂x‘
þ Γmi‘Γ ‘mj �
1
2
Γmij
∂ ‘ngð Þ
∂xm
ð5:4:3Þ
If g < 0 it is enough to change g for �g in the expression (5.4.3). The
permutation of indexes j $ i leads to
Rji ¼ 1
2
∂2 ‘ngð Þ
∂xi∂xj
� ∂Γ
‘
ji
∂x‘
þ Γmj‘Γ ‘mi �
1
2
Γmji
∂ ‘ngð Þ
∂xm
As the Christoffel symbols are symmetric and the order of differentiation in the
first term of the previous expression is independent of the sequence in which it is
carried out, it is concluded that the Ricci tensor Rij is symmetric, so it has
N
2
N þ 1ð Þ
distinct components.
The contractions that can be carried out in tensor R‘ijk are: R
‘
‘jk,R
‘
i‘k,R
‘
ij‘. Con-
sidering the antisymmetry of curvature tensor R ‘ijk ¼ �R ‘ikj and with k ¼ ‘ the result
isR ‘ij‘ ¼ �R ‘i‘j, whereby Rij ¼ �R ‘i‘j. The contraction R‘i‘k generates the Ricci tensor
Rij with sign changed, then it is enough to consider only the contraction R
‘
ij‘ to
obtain tensor Rij, which contains components independent of R
‘
ijk in the more
adequate form of a symmetric tensor.
The contraction of tensor R‘ijk in the indexes i ¼ ‘ is given by
R ‘‘jk ¼
∂Γ ‘‘k
∂xj
� ∂Γ
‘
‘j
∂xk
þ Γm‘kΓ ‘mj � Γm‘jΓ ‘mk
and with the expressions
Γ ‘‘k ¼
∂ ‘n
ffiffiffi
g
p� �
∂xk
Γ ‘‘j ¼
∂ ‘n
ffiffiffi
g
p� �
∂xj
the result is
R ‘‘jk ¼
∂
∂xj
∂ ‘n
ffiffiffi
g
p� �
∂xk
 �
� ∂
∂xk
∂ ‘n
ffiffiffi
g
p� �
∂xj
 �
þ Γm‘kΓ ‘mj � Γm‘jΓ ‘mk
The permutation of indexes ‘ $ m in the last term and the symmetry of the
Christoffel symbols allow writing
R ‘‘jk ¼
∂
∂xj
∂ ‘n
ffiffiffi
g
p� �
∂xk
 �
� ∂
∂xk
∂ ‘n
ffiffiffi
g
p� �
∂xj
 �
þ Γm‘kΓ ‘mj � Γ ‘mjΓm‘k
and as
252 5 Riemann Spaces
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∂
∂xj
∂ ‘n
ffiffiffi
g
p� �
∂xk
 �
¼ ∂
∂xk
∂ ‘n
ffiffiffi
g
p� �
∂xj
 �
then
R ‘‘jk ¼ Rjk ¼ 0
It is concluded that the contraction of the Riemann–Christoffel curvature tensor
R‘ijk in the indexes ‘ ¼ i generates the null tensor.
5.4.2 Divergence of the Ricci Tensor with Variance
Ricci (0, 2)
The calculation of the divergence of tensor R‘ijk is carried out considering the second
Bianchi identity
∂‘R
‘
ijk þ ∂jR ‘ik‘ þ ∂kR ‘i‘j ¼ 0
in which the contraction of the indexes ‘ ¼ k provides
∂‘R
k
ijk þ ∂jRkik‘ þ ∂kRki‘j ¼ 0 ) ∂‘Rij þ ∂jRi‘ þ divRki‘j ¼ 0
whereby
divRki‘j ¼ � ∂‘Rij þ ∂jRi‘
� �
and with the ordination of the indexes
divR ‘ijk ¼ � ∂jRik þ ∂kRij
� � ð5:4:4Þ
5.4.3 Bianchi Identity for the Ricci Tensor
with Variance (0, 2)
An identity analogous to the second Bianchi identity can be obtained for the Ricci
tensor. Rewriting expression (5.2.15)
∂pR
‘
ikj þ ∂jR ‘ikp þ ∂kR ‘ipj ¼ 0
5.4 Ricci Tensor and Scalar Curvature 253
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and with the relations
g ‘i Rkj ¼ R ‘ikj g ‘i Rkp ¼ R ‘ikp g ‘i Rpj ¼ R ‘ipj
it follows that
g ‘i ∂pRkj ¼ ∂pR ‘ikj g ‘i ∂iRkp ¼ ∂iR ‘ikp g ‘i ∂kRpj ¼ ∂kR ‘ipj
The sum of these three expressions provides
g ‘i ∂pRkj þ ∂iRkp þ ∂kRpj
� � ¼ ∂pR ‘ikj þ ∂iR ‘ikp þ ∂kR ‘ipj
As the term to the right is the second Ricci identity it results in
∂pRkj þ ∂iRkp þ ∂kRpj ¼ 0
The changes of the indexes j ! i, k ! j, p ! k allow the ordination of the same,
then
∂kRij þ ∂iRjk þ ∂jRki ¼ 0 ð5:4:5Þ
that is called Bianchi identity for the Ricci tensor of covariant components.
5.4.4 Scalar Curvature
The multiplying of the Ricci tensor Rij by the conjugate metric tensor g
ij provides
R ¼ gijRij ð5:4:6Þ
that defines the scalar curvature, which is the trace of the Ricci tensor, also called
Ricci curvature or invariant curvature of the Riemann space EN.
5.4.5 Geometric Interpretation of the Ricci Tensor
with Variance (0, 2)
Let the Riemann curvature
K xi; u; v
� � ¼ Rk‘mnukv‘umvn
gkmg‘n � gkngm‘ð Þukv‘umvn
254 5 Riemann Spaces
where u, v are orthogonal unit vectors, the result thereof is
gkmg‘nu
kv‘umvn ¼ gkmukum
� �
g‘nv
‘vn
� �
gkngm‘u
kv‘umvn ¼ gknukvn
� �
gm‘u
mv‘
� �
but
gkmu
kum ¼ g‘nv‘vn ¼ 1 gknukvn ¼ gm‘umv‘ ¼ 0
then
Kuv ¼ K xi; u; v
� � ¼ Rk‘mnukv‘umvn
1� 1� 0 ¼ Rk‘mnu
kv‘umvn
where the notation Kuv is adopted by convenience of graphic representation. If the
unit vectors u, v are linearly dependent, the result is K ¼ 0.
The summation of all the N components of vector u is given by
XN
vj¼1
Kuv ¼
XN
vj¼1
Rk‘mnu
kv‘umvn ¼ ukum
XN
vj¼1
Rk‘mnv
‘vn
but
XN
vj¼1
v‘vn ¼ g‘n
whereby the contraction R‘i‘j generates the Ricci tensor Rij with the signchanged,
then
XN
vj¼1
Kuv ¼ �ukumg‘nRk‘mn ¼ �ukumRnkmn ¼ �ukumRkm
Putting
Ku ¼
XN
vj¼1
Kuv ¼ �ukumRkm ð5:4:7Þ
where Ku is the sum of the Riemann curvature for the space EN determined by the
components of vector u and each N � 1ð Þ directions which are mutually orthogonal
to them. This expression is independent of these directions and defines the mean
curvature of EN in the direction of this vector.
5.4 Ricci Tensor and Scalar Curvature 255
In expression (5.4.7) when carrying out the summation on the N directions
mutually orthogonal, it follows that
XN
ui¼1
Ku ¼ �
XN
ui¼1
ukumRkm
XN
ui¼1
ukum ¼ gkm
XN
ui¼1
Ku ¼ �gkmRkm ¼ �R ð5:4:8Þ
Expression (5.4.8) shows that the sum of the mean curvatures in the Riemann
space EN for mutually orthogonal directions are independent of the directions
defined by the vectors u, v, being equal to the scalar curvature.
5.4.6 Eigenvectors of the Ricci Tensor with Variance (0, 2)
The Ricci tensor Rij is symmetric and has in each point of the Riemann space EN a
system of linearly independent equations that define principal directions
(eigenvectors).
Let the Riemann curvature
K xi; u; v
� � ¼ Rk‘mnukv‘umvn
gkmg‘n � gkngm‘ð Þukv‘umvn
where the vectors are orthogonal and only v is a unit vector, so
gkmg‘nu
kv‘umvn ¼ gkmukum
� �
g‘nv
‘vn
� �
gkngm‘u
kv‘umvn ¼ gknukvn
� �
gm‘u
mv‘
� � ¼ 0
then
K xi; u; v
� � ¼ Rk‘mnukv‘umvn
gkmu
kumð Þ g‘nv‘vnð Þ
but as v is a unit vector the result is
g‘nv
‘vn ¼ 1 ) v‘vn ¼ g‘n
256 5 Riemann Spaces
whereby
K xi; u; v
� � ¼ �Rk‘mng‘nukum
gkmu
kum
thereof
Ku ¼ K xi; u; v
� � ¼ �Rkmukum
gkmu
kum
ð5:4:9Þ
is the normalized mean curvature, where the index indicates that u is not unit
vector.
The calculation of the eigenvalues is carried out by means of the equations
system
Rkm þ Kugkmð Þukum ¼ 0
with extreme values given by the condition
∂
∂uk
Rkm þ Kugkmð Þukum
� � ¼ 0
which developed stays
2 Rkm þ Kugkmð Þum þ
∂Rkm
∂uk
ukum þ ∂Ku
∂uk
gkmu
kum ¼ 0
and as the Ricci tensor Rij does not depend on vector u
k the result is
2 Rkm þ Kugkmð Þum þ
∂Ku
∂uk
gkmu
kum ¼ 0
For the extreme values of Ku the result is
∂Ku
∂xk ¼ 0, whereby the equations system
Rkm þ Kugkmð Þum ¼ 0
allows determining the principal directions (eigenvectors) of the Ricci tensor Rij.
5.4.7 Ricci Tensor with Variance (1, 1)
The Ricci tensor in terms of its mixed components is given by
Rij ¼ gimRmj ð5:4:10Þ
5.4 Ricci Tensor and Scalar Curvature 257
An important expression that relates the Ricci tensor with variance (1, 1) with
the derivative of the scalar curvature can be obtained by means of the second
Bianchi identity
∂pR
‘
ijk þ ∂jR ‘ikp þ ∂kR ‘ipj ¼ 0
where with the antisymmetry R ‘ikp ¼ �R ‘ipk the result is
∂pR
‘
ijk � ∂jR ‘ipk þ ∂kR ‘ipj ¼ 0
The contraction of these tensors in indexes ‘ ¼ k provides
∂pRij � ∂jRip þ ∂kRkipj ¼ 0
Multiplying by gip it follows that
gip∂pRij � gip∂jRip þ gip∂kRkipj ¼ 0
∂pg
ipRij � ∂jgipRip þ ∂kgipRkipj ¼ 0
∂pR
p
j �
∂R
∂xj
þ ∂kRkj ¼ 0 )
∂R
∂xj
¼ ∂pRpj þ ∂kRkj
The change of the dummy indexes p ! k provides
∂R
∂xj
¼ 2∂kRkj
whereby
∂kR
k
j ¼
1
2
∂R
∂xj
ð5:4:11Þ
For the Riemann space EN, with N > 2, multiplying expression (5.4.5) by g
ij the
result is
gij∂kRij þ gij∂iRjk þ gij∂jRki ¼ 0 ) ∂kgijRij þ ∂igijRjk þ ∂jgijRki ¼ 0
and having curvature R a scalar function at its partial derivative is equal to its
covariant derivative, then
∂R
∂xk
þ ∂iR ik þ ∂jR jk ¼ 0
258 5 Riemann Spaces
and with the change of indexes j ! i the result is
∂R
∂xk
þ 2∂iR ik ¼ 0
and with
∂iR
i
k ¼
1
2
∂R
∂xk
it follows that
∂R
∂xk
þ 2 � 1
2
∂R
∂xk
¼ 0 ) ∂R
∂xk
¼ 0
then the scalar curvature is constant for this kind of space. The purpose of the
suppositionN > 2will be clarified by expression (5.6.10), obtained when analyzing
the scalar curvature in the Riemann space E2.
Exercise 5.1 For the tensorial expression T ij ¼ Rij þ δ ij αRþ βð Þ, where α, β are
scalars, calculate the value of α so that the covariant derivative ∂iT ij is null.
The null covariant derivative ∂iT ij is given by
∂iT
i
j ¼ ∂iR ij þ ∂i δ ij αRþ βð Þ
h i
having ∂iδ ij ¼ 0 it follows that
∂iT
i
j ¼ ∂iR ij þ α∂iR ¼ 0
With the expression (5.4.11)
∂iR
i
j ¼
1
2
∂R
∂xj
) ∂iT ij ¼
1
2
þ α
� �
∂R
∂xj
¼ 0
for ∂iR ¼ ∂R∂xj, and as this derivative assumes any values the result is α ¼ �12.
5.4.8 Notations
In Table 5.2, in which the Tulio Levi-Civita notation was inserted, there is a
compilation of the evolution of the notation for the Riemann–Christoffel curvature
tensors and for the Ricci tensor. The notations that make use of (,) or (;) seek to
5.4 Ricci Tensor and Scalar Curvature 259
indicate the properties of symmetry and antisymmetry of the Riemann–Christoffel
tensors. In the case of using (.) it indicates the index, or the position and the index
that will be lowered or raised. The only difference between the two notations of
Christoffel and Bianchi is the change of the point and comma (;) for the comma (,).
Currently these two forms of spelling were abandoned. It is stressed that several
authors have opted for different positioning of the indexes. The Weyl notation, with
the change of the letter F for R (Riemann), was the one that became consecrated in
the current literature.
Exercise 5.2 In a coordinates system let Γ ijk ¼ δ ij ∂ϕ∂xk þ δ ik ∂ψ∂xj, where ϕ,ψ are
functions of position. Calculate: (a) Rijk‘; (b) Rjk for ψ ¼ �‘n aixið Þ.
(a) Substituting the expression
Γ ijk ¼ δ ij
∂ϕ
∂xk
þ δ ik
∂ψ
∂xj
in the expression of the Riemann–Christoffel curvature tensor
Rijk‘ ¼
∂Γ ij‘
∂xk
� ∂Γ
i
jk
∂x‘
þ Γ irkΓ rj‘ � Γ ir‘Γ rjk
Table 5.2 Notations for the Riemann–Christoffel curvature tensors and Ricci tensor
Author
Riemann–Christoffel curvature tensor
Ricci tensor
Mixed variance
components (1, 3)
Covariant
components (0, 4)
Brillouin Rij; k‘ Rij, k‘ Rj‘ ¼
X
m
Rmj,m‘
Appe-Thiry Ri
 j k‘ Rijk‘ Rjk ¼
X
m
Rm
jkm
Weyl Fijk‘ Fijk‘ Rj‘ ¼
X
m
Fmjm‘
Eddington-
Becquerel
Bij k‘ Bjk‘i Gjk ¼
X
m
Bmjmk
Galbrun Rij‘k Rij‘k Rjk ¼
X
m
Rmjmk
Juvet R
 ij 
 ‘k Rji‘k Rjk ¼
X
m
R
mj
mk
Cartan Rij;‘k Rji,‘k Rj‘ ¼
X
m
Rmj‘m
Christoffel and
Bianchi
( ji; k‘) ( ji, k‘) –
Levi-Civita {ji, k‘} ( ji, k‘) αj‘
260 5 Riemann Spaces
it follows that
Rijk‘ ¼
∂
∂xk
δ ij
∂ϕ
∂x‘
þ δ i‘
∂ψ
∂xj
� �
� ∂
∂x‘
δ ij
∂ϕ
∂xk
þ δ ik
∂ψ
∂xj
� �
þ δ ij
∂ϕ
∂xk
þ δ ik
∂ψ
∂xr
� �
δ rj
∂ϕ
∂x‘
þ δ r‘
∂ψ
∂xj
� �
� δ ir
∂ϕ
∂x‘
þ δ i‘
∂ψ
∂xr
� �
δ rj
∂ϕ
∂xk
þ δ rk
∂ψ
∂xj
� �
¼ δ ij
∂2ϕ
∂xk∂x‘
þ δ i‘
∂2ψ
∂xk∂xj
� δ ij
∂2ϕ
∂x‘∂xk
� δ ik
∂2ψ
∂x‘∂xj
þ δ ij δ rj
∂ϕ
∂xk
∂ϕ
∂x‘
þ δ ij δ r‘
∂ϕ
∂x‘
∂ψ
∂xj
þδ ikδ rj
∂ψ
∂xr
∂ϕ
∂x‘
þ δ ikδ r‘
∂ψ
∂xr
∂ψ
∂xj
� δ irδ rj
∂ϕ
∂x‘
∂ϕ
∂xk
� δ irδ rk
∂ϕ
∂x‘
∂ψ
∂xj
� δ i‘δ rj
∂ψ
∂xr
∂ϕ
∂xk
�δ i‘δ rk
∂ψ
∂xr
∂ψ
∂xj
R ijk‘ ¼ δ ij
∂ϕ
∂xk
∂ϕ
∂x‘
þ δ ik
∂ψ
∂xj
∂ψ
∂x‘
þ δ ik
∂ϕ
∂x‘
∂ψ
∂xj
þ δ i‘
∂ϕ
∂xk
∂ψ
∂xj
� δ ij
∂ϕ
∂x‘
∂ϕ
∂xk
�δ i‘
∂ψ
∂xj
∂ψ
∂xk
� δ i‘
∂ϕ
∂xk
∂ψ
∂xj
� δ ik
∂ϕ
∂x‘
∂ψ
∂xj
þ δ ij
∂2ϕ
∂x‘∂xk
þ δ i‘
∂2ψ
∂xj∂xk
�δ ij
∂2ϕ
∂xk∂x‘
� δ ik
∂2ψ
∂xj∂x‘
Rijk‘ ¼ δ ik
∂ψ
∂xj
∂ψ
∂x‘
� ∂
2ψ
∂xj∂x‘
 !
� δ i‘
∂ψ
∂xj
∂ψ
∂xk
� ∂
2ψ
∂xj∂xk
 !
then Rijk‘ only depends on the function ψ .
(b) For ψ ¼ �‘n aixið Þ the partial derivatives result
∂ψ
∂xj
¼ � aj
aixi
) ∂
2ψ
∂xj∂x‘
¼ aja‘
aixið Þ2
∂ψ
∂x‘
¼ � a‘
aixi
) ∂ψ
∂xj
∂ψ
∂x‘
¼ aja‘
aixið Þ2
and substituting this derivatives in the expression obtained in item (a) it follows
that
5.4 Ricci Tensor and Scalar Curvature 261
Rijk‘ ¼ δ ik
∂ψ
∂xj
∂ψ
∂x‘
� ∂
2ψ
∂xj∂x‘
 !
� δ i‘
∂ψ
∂xj
∂ψ
∂xk
� ∂
2ψ
∂xj∂xk
 !
Rijk‘ ¼ δ ik
aja‘
aixið Þ2
� aja‘
aixið Þ2
" #
� δ i‘
ajak
aixið Þ2
� ajak
aixið Þ2
" #
¼ 0
whereby
Rijki ¼ Rjk ¼ 0 Q:E:D:
5.5 Einstein Tensor
The tensor Rijk‘, the second Bianchi identity, the Ricci tensor Rij and the scalar
curvature R allow obtaining a second-order tensor with peculiar characteristics. Let
the second Bianchi identity
∂mRijk‘ þ ∂kRij‘m þ ∂‘Rijmk ¼ 0
and with the antisymmetry of the Riemann–Christoffel curvaturetensor Rijk‘
∂mRijk‘ � ∂kRijm‘ � ∂‘Rjimk ¼ 0
and multiplying by gi‘ and gjk it follows that
gi‘gjk∂mRijk‘ � gi‘gjk∂kRijm‘ � gi‘gjk∂‘Rjimk ¼ 0
gjk∂mR
‘
jk‘ � gjk∂kR ‘jm‘ � gi‘∂‘Rkimk ¼ 0
whereby in terms of the Ricci tensor
gjk∂mRjk � gjk∂kRjm � gi‘∂‘Rim ¼ 0
The change of the dummy index ‘ ! k in the last term provides
gjk∂mRjk � gjk∂kRjm � gik∂kRim ¼ 0 ) ∂mRjkjk � ∂kRjkjm � ∂kRikim ¼ 0
The contractions of the curvature tensors provide
∂mR� ∂kRkm � ∂kRkm ¼ 0 ) ∂mR ¼ 2∂kRkm
262 5 Riemann Spaces
whereby
∂kR
k
m ¼
1
2
∂mR ð5:5:1Þ
is the divergence of a tensor, which can be written under the form
∂k R
k
m �
1
2
δ kmR
� �
¼ 0 ð5:5:2Þ
where the terms in parenthesis define the Einstein tensor with variance (1, 1)
Gkm ¼ Rkm �
1
2
δ kmR ð5:5:3Þ
The Einstein tensor can be written as a function of its covariant components, so
Gij ¼ gikGkj ¼ gik Rkj �
1
2
δ kj R
� �
ð5:5:4Þ
thus
Gij ¼ Rij � 1
2
gijR ð5:5:5Þ
By means of this expression it is verified that the Einstein tensor is generated
only by the metric tensor and the Ricci tensor. As Rij and gij are two symmetric
tensors then Einstein tensor is symmetric. For the contravariant components of this
tensor the result is
Gij ¼ Rij � 1
2
gijR ð5:5:6Þ
The divergence of the Einstein tensor is given by
∂iG
j
i ¼ ∂iR ij � δ ji
1
2
∂iR ¼ ∂iR ij �
1
2
∂jR
but
∂iR
i
j ¼
1
2
∂jR
then
∂iG
i
j ¼ 0 ð5:5:7Þ
5.5 Einstein Tensor 263
Thus for any Riemann space the divergence of the Einstein tensor is null, and
with the contraction of this tensor it follows that
Gii ¼ Rii �
1
2
δ ii R ¼ R�
1
2
NR
G ¼ �1
2
N � 2ð ÞR ð5:5:8Þ
For the Riemann space E2 it is verified that G ¼ 0.
Exercise 5.3 Show that the tensor of the kind T ij ¼ Rij þ δ ij m, being m a scalar
function, has the characteristics of an Einstein tensor.
The divergence of this tensor given by ∂jT ij ¼ 0 stays
∂jT
i
j ¼ ∂jR ij þ δ ij∂jm ¼ ∂j R ij þ m
� �
¼ 0
and with expression (5.4.11)
∂jR
i
j ¼
1
2
∂jR
substituted in this expression
∂jT
i
j ¼ ∂j
1
2
Rþ m
� �
¼ 0 ) 1
2
Rþ m ¼ k1 ) m ¼ k1 � 1
2
R
where k1 is a constant. The substitution of this expression in the expression of
tensor Tij provides
T ij ¼ Rij � δ ij
1
2
Rþ k2
� �
where k2 ¼ �k1.
Thus this tensor has the same characteristics of the Einstein tensor defined by
expression (5.5.3).
5.6 Particular Cases of Riemann Spaces
Some kinds of Riemann spaces will be analyzed in this item with specific charac-
teristics that make them important: the Riemann space E2, the Riemann space with
constant curvature, the Minkowski space, and the conformal space.
264 5 Riemann Spaces
5.6.1 Riemann Space E2
In the Riemann space E2 the Ricci tensor Rij is defined by its components
Rij ¼
R11 R12
R21 R22
" #
as R12 ¼ R21 and the metric tensor in matrix form is given by
gij ¼
g11 g12
g21 g22
" #
where g12 ¼ g21.
The Ricci tensor written in terms of the Riemann–Christoffel curvature tensor
with variance (0, 4), and considering the symmetry and the metric tensor is given by
Rij ¼ gkpRpijk ¼ gpkRipkj
and the development provides
Rij ¼ g11Ri11j þ g12Ri12j þ g21Ri21j þ g22Ri22j
whereby the result for component R11 is
R11 ¼ g11R1111 þ g12R1121 þ g21R1211 þ g22R1221
As the tensor Rpijk is antisymmetric in the first two and the last two indexes, i.e.,
Rpijk ¼ �Ripjk and Rpijk ¼ �Rpikj it follows that
R11 ¼ 0þ 0þ 0þ g22R1221
Let g ¼ detgij and G22 the cofactor of g22:
g22 ¼ G
22
g
¼ g11
g
whereby
R11 ¼ g11
g
�R1212ð Þ ) R11
g11
¼ �R1212
g
Proceeding in an analogous way for component R22:
5.6 Particular Cases of Riemann Spaces 265
R22 ¼ g11R2112 þ g12R2122 þ g21R2212 þ g22R2222
R22 ¼ g11R2112 þ 0þ 0þ 0
R22 ¼ �g11R1212
g11 ¼ G
11
g
¼ g22
g
R22 ¼ �g22
g
R1212
whereby
R22
g22
¼ �R1212
g
For component R12, it follows that
R12 ¼ g11R1112 þ g12R1122 þ g21R1212 þ g22R1222 ¼ 0þ 0þ g21R1212 þ 0
R12 ¼ g21R1212
g21 ¼ G
21
g
¼ g12
g
R12 ¼ �g12
g
R1212
thus
R12
g12
¼ �R1212
g
and with the symmetries Rij ¼ Rji and gij ¼ gji the result for component R21 is
R21
g21
¼ �R1212
g
The analysis developed shows that
K ¼ R11
g11
¼ R22
g22
¼ R12
g12
¼ R21
g21
¼ �R1212
g
These equalities indicate that in the Riemann space E2 the components of the
Ricci tensor Rij are proportional to the components of the metric tensor gij and to its
derivatives, and are independent of the directions considered. It is verified that the
Riemann curvature does not vary with the orientation considered, then all the points
266 5 Riemann Spaces
of the space E2 are isotropic. This, in general, is not valid for spaces with dimension
N > 2. The scalar K in Riemann space E2 is called Gauß curvature.
This analysis allows writing the components of the Ricci tensor as a function of
the component R1212 and of the metric tensor, thus
Rij ¼ �R1212
g
gij ð5:6:1Þ
5.6.2 Gauß Curvature
Expression (5.6.1) is valid only for the Riemann space E2. The knowledge of the
properties of the surfaces in the Euclidian space E3 is not useful for understanding
the properties of the Riemann spaces EN, with N > 3. For N ¼ 2 several simplifi-
cations are admitted in the formulation of the expression of Rij, so the conclusions
obtained for the Riemann space E2 cannot be generalized for the spaces of dimen-
sions N > 3.
The scalar curvature allows expressing the Riemann–Christoffel tensor Rpijk as a
function of the components of the metric tensor.
With the non-null components R1212, ¼ �R2121, ¼ �R1221 ¼ R2112, and the
expression of the scalar curvature it follows that
R ¼ gijRij ¼ �gijgij
R1212
g
¼ �δ ii
R1212
g
¼ �2
g
R1212 ) R1212 ¼ �R
2
g
and the development provides
R1212 ¼ �R
2
g11 g12
g21 g22
				 				 ¼ �R2 g11g22 � g12g21ð Þ
The other non-null components are obtained by means of the indexes in this
expression, and considering the symmetry of tensor Rpijk it follows that
R2121 ¼ �R
2
g22g11 � g21g12ð Þ
R1221 ¼ �R
2
g12g21 � g11g22ð Þ
R2121 ¼ �R
2
g21g12 � g22g11ð Þ
5.6 Particular Cases of Riemann Spaces 267
then
Rijk‘ ¼ �R
2
gikgj‘ � gi‘gjk
� �
ð5:6:2Þ
or
Rijk‘ ¼ �K gikgj‘ � gi‘gjk
� �
ð5:6:3Þ
The Gauß curvature, that in general depends on the coordinates of the point
considered, is determined by
K ¼ 1
2
R ð5:6:4Þ
that can be obtained as a function of the Riemann–Christoffel curvature tensor with
variance (0, 4), and with the Ricci pseudotensor for the Riemann space E2
εij ¼ ffiffiffigp eij εij ¼ eijffiffiffi
g
p
and with the expression
K ¼ R1212
g
then
Rijk‘ ¼ Kεijεk‘ ð5:6:5Þ
The multiplication of both members of this expression by εijεk‘ provides
εijεk‘Rijk‘ ¼ Kεijεk‘εijεk‘
and as
εijε
ij ¼ δ ii ¼ 2
thus
K ¼ 1
4
Rijk‘ε
ijεk‘ ð5:6:6Þ
this expression shows that the Gauß curvature is an invariant.
268 5 Riemann Spaces
5.6.3 Component R1212 in Orthogonal Coordinate Systems
For the orthogonal coordinate systems in the Riemann space EN expression (5.2.24)
provides the component
R1212 ¼ 1
2
2
∂2g12
∂x1∂x2
� ∂
2
g11
∂x2∂x2
� ∂
2
g22
∂x1∂x1
 !
þ gq‘ Γ q12Γ ‘12 � Γ q11Γ ‘22
� �
or more explicitly
R1212 ¼�1
2
∂2g11
∂x2∂x2
þ ∂
2
g22
∂x1∂x1
 !
þg11 Γ112Γ112�Γ111Γ122
� �þg22 Γ212Γ212�Γ211Γ222� �
The Christoffel symbols for these coordinates systems are given by
– i ¼ j ¼ k ) Γ kij ¼ Γ iii ¼ 12gii
∂gii
∂xj )
Γ111 ¼
1
2g11
∂g11
∂x1
Γ222 ¼
1
2g22
∂g22
∂x2
8>>><>>>:
– i ¼ j 6¼ k ) Γ kij ¼ Γ kii ¼ � 12gkk
∂gii
∂xk )
Γ211 ¼ �
1
2g22
∂g11
∂x2
Γ122 ¼ �
1
2g11
∂g22
∂x1
8>>><>>>:
– i ¼ k 6¼ j ) Γ kij ¼ Γ iij ¼ 12gii
∂gii
∂xj )
Γ112 ¼
1
2g11
∂g11
∂x2
Γ212 ¼
1
2g22
∂g22
∂x1
8>>><>>>:
– For i 6¼ j, j 6¼ k, i 6¼ k it results in Γij,k ¼ 0
so
R1212 ¼ �1
2
∂2g11
∂x2∂x2
þ ∂
2
g22
∂x1∂x1
 !
þ 1
4g11
∂g11
∂x2
� �2
þ ∂g11
∂x1
∂g22
∂x1
" #
þ 1
4g22
∂g22
∂x1
� �2
þ ∂g11
∂x2
∂g22
∂x2
" #
¼ � 1
2
ffiffiffiffiffiffiffiffiffiffiffiffi
g11g22
p ∂
∂x1
1ffiffiffiffiffiffiffiffiffiffiffiffi
g11g22
p ∂g22
∂x1
� �
þ ∂
∂x2
1ffiffiffiffiffiffiffiffiffiffiffiffi
g11g22
p ∂g11
∂x2
� �
 �
5.6 Particular Cases of Riemann Spaces 269
or
R1212 ¼ � 12
ffiffiffi
g
p ∂
∂x1
1ffiffiffi
g
p ∂g22
∂x1
� �
þ ∂
∂x2
1ffiffiffi
g
p ∂g11
∂x2
� �
 �
ð5:6:7Þ
Exercise 5.4 Calculate the components of tensors Rijk‘, Rij, and the Gauß curvature
for the space E2 defined by the fundamental form ds
2 ¼ c2 dx1ð Þ 2 � f 2 tð Þ dx2ð Þ2
where c2 is a constant.
The metric tensor and conjugated metric tensor are given, respectively, by
gij ¼
c2 0
0 �f 2 tð Þ
" #
gij ¼ c
�2 0
0 �f�2 tð Þ
" #
then
g ¼ c2f 2i2 ) ffiffiffigp ¼ cf i
where i2 ¼ �1 is the imaginary number and with expression (5.6.8)
R1212 ¼ � 1
2
ffiffiffi
g
p ∂
∂x1
1ffiffiffi
g
p ∂g22
∂x1
� �
þ ∂
∂x2
1ffiffiffi
g
p ∂g11
∂x2
� �
 �
it follows that
R1212 ¼ � 1
2cf i
∂
∂x1
1
cf i
∂g22
∂x1
� �
¼ � 1
2cf i
∂
∂x1
1
cf i
� �2f _f� �
 � ¼ 1
2cf i
∂
∂x1
2 _f
ci
� �
¼
€f
c2f i2
¼ �
€f
c2f
For the components of the Ricci tensor it follows that
Rij ¼ gpkRipkj
R11 ¼ g22R1212 ¼ � 1
f 2
�
€f
c2f
� �
¼
€f
c2f 3
R22 ¼ g11R1212 ¼ 1
c2
�
€f
c2f
� �
¼ �
€f
c4f
R12 ¼ R21 ¼ g12R1212 ¼ 0
and for the Gauß curvature it results in
K ¼ R1212
g
¼
� €fc2f
c2f 2i2
¼
€f
c4f 3
270 5 Riemann Spaces
5.6.4 Einstein Tensor
For the particular case in which the metric, the metric tensor, and its conjugated
tensor are given, respectively, by
ds2 ¼ h x1; x2� � dx1� �2 þ h x1; x2� � dx2� �2
gij ¼
h 0
0 h
" #
gij ¼
1
h
0
0
1
h
2664
3775
where h x1; x2ð Þ > 0 is a function of the coordinates, g ¼ detgij ¼ h2, and the Ricci
tensor is expressed by
Rij ¼ gpkRipkj ¼ g11Ri11j þ g12Ri12j þ g21Ri21j þ g22Ri22j
then
Rij ¼ 1
h
Ri11j þ Ri22j
� �
Developing this expression and with the symmetry of tensor Ripkj it follows that
R11 ¼ 1
h
R1111 þ R1221ð Þ ¼ 1
h
R1221 R22 ¼ 1
h
R2112 þ R2222ð Þ ¼ 1
h
R2112
R12 ¼ 1
h
R1112 þ R1222ð Þ ¼ 0 R21 ¼ 1
h
R2111 þ R2221ð Þ ¼ 0
Let the scalar curvature
R ¼ gijRij ¼ g11R11 þ g12R12 þ g21R21 þ g22R22 ¼ g11R11 þ 0þ 0þ g22R22
¼ g11R11 þ g22R22
and with the components of the Ricci tensor as a function of the components of
tensor Ripkj it follows that
R ¼ 1
h
1
h
R1221 þ 1
h
1
h
R2112
As Ripkj ¼ Rpijk it results for the scalar curvature
5.6 Particular Cases of Riemann Spaces 271
R ¼ 1
h2
R1221 þ R1221ð Þ ¼ 2
h2
R1221
then
R1221 ¼ h
2
2
R
and with the substitution of this expression in the expressions of the components of
the Ricci tensor it follows that
R11 ¼ 1
h
h2
2
R ¼ h
2
R ¼ R
2
g11
R22
1
h
h2
2
R ¼ h
2
R ¼ R
2
g22
R12 ¼ R21 ¼ 0
These expressions allow relating the Ricci tensor with the scalar curvature and
with the metric tensor, thus
Rij ¼ R
2
gij ð5:6:8Þ
and with the definition of the scalar curvature given by expression (5.4.6) and with
the previous expression it follows
R ¼ gijRij ¼ gijgij
R
2
¼ δ ii
R
2
¼ NR
2
or
R 1� N
2
� �
¼ 0 ð5:6:9Þ
then for the Riemann space E2 it is verified that Rij ¼ R ¼ 0.
Consider the Einstein tensor given by its covariant components
Gij ¼ Rij � 1
2
gijR ¼ �Kgij �
1
2
gijR ¼ �Kgij �
1
2
gij �2Kð Þ ¼ 0
then the tensor Gij is null for the Riemann space E2.
272 5 Riemann Spaces
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5.6.5 Riemann Space with Constant Curvature
The Riemann curvature in point xi2EN , in general, depends on this point in which it
is defined and the vectors u and v that establish the plane π with respect to which it
is calculated. It is admitted that this dependency does not exist, i.e., the space is
isotropic, then the relation of the isotropy of the space with the Riemann curvature
is established by the following theorem.
Schur Theorem
If all the points of a neighborhood in the Riemann space EN, being N > 2, are
isotropic, then the curvature K is constant in all this neighborhood.
To prove the validity of this theorem, let expression (5.3.13) be rewritten as
Rijk‘ ¼ Gijk‘K ð5:6:10Þ
with
Gijk‘ ¼ gikgj‘ � gi‘gjk
� �
6¼ 0
valid in the neighborhood of point xm of Riemann space EN.
The covariant derivative of expression (5.6.11) with respect to variable xm is
given by
∂mRijk‘ ¼ Gijk‘∂mK ð5:6:11Þ
with ∂mGijk‘ ¼ 0, because, in general, ∂gij∂xm ¼ 0.
With the permutation of indexes in the expression (5.6.12)
∂kRij‘m ¼ Gij‘m∂kK ð5:6:12Þ
∂‘Rijmk ¼ Gijmk∂‘K ð5:6:13Þ
The sum of expressions (5.6.12)–(5.6.14) provides
∂mRijk‘ þ ∂kRij‘m þ ∂‘Rijmk ¼ Gijk‘∂mK þ Gij‘m∂kK þ Gijmk∂‘K
but the left side of expression is the second Bianchi identity thus
Gijk‘∂mK þ Gij‘m∂kK þ Gijmk∂‘K ¼ 0
and multiplying the terms of this expression by gikgj‘ it follows
5.6 Particular Cases of Riemann Spaces 273
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gikgj‘Gijk‘∂mK ¼ gikgj‘ gikgj‘ � gi‘gjk
� �
¼ δ kk δ ‘‘ � δ k‘ δ ‘k ¼ N2 � N
gikgj‘Gij‘m∂kK ¼ gikgj‘ gi‘gjm � gimgj‘
� �
¼ δ k‘ δ ‘m � δ kmδ ‘‘ ¼ δ km � Nδ km
gikgj‘Gijmk∂‘K ¼ gikgj‘ gimgjk � gikgjm
� �
¼ δ kmδ ‘k � δ kk δ ‘m ¼ δ ‘m � Nδ ‘m
The sum of these three terms provides
N2 � N� �∂mK þ δ km � Nδ km� �∂kK þ δ ‘m � Nδ ‘m� �∂‘K ¼ 0
it follows that
N2 � N� �∂mK þ 1� Nð Þ∂mK þ 1� Nð Þ∂mK ¼ 0
whereby
N2 � N� �þ 2 1� Nð Þ� �∂mK ¼ 0 ð5:6:14Þ
For N > 2 this expression is null only if ∂mK ¼ 0, and as xm is an arbitrary
coordinate it is concluded that K is constant in the neighborhood of this point in the
Riemann space EN, which proves the Schur theorem. Expression (5.3.13), where
K is a constant is the necessary and sufficient condition so that the curvature of the
Riemann space EN is independent of the orientation considered.
5.6.6 Isotropy
Another characteristic of this type of space is related with a scalar curvature. Let
expression (5.3.13) be rewritten as
Rijk‘ ¼ K gikgj‘ � gi‘gjk
� �
and multiplied by g‘i
Rjk ¼ g‘iRijk‘ ¼ Kg‘i gikgj‘ � gi‘gjk
� �
¼ K δ ‘k gj‘ � δ ‘‘gjk
� �
¼ K gjk � Ngjk
� �
then
Rjk ¼ K 1� Nð Þgjk ð5:6:15Þ
274 5 Riemann Spaces
For the scalar curvature it follows that
Rkk ¼ gkjRjk ¼ gkjK 1� Nð Þgjk ¼ K 1� Nð Þδ kk
whereby
R ¼ K 1� Nð ÞN ð5:6:16Þ
This formulation shows that in the Riemann space E2 the tensor Rijk‘ leads to the
Gauß curvature K, which is the reason for adopting the denomination curvature
tensor by extension of this particular case for Riemann spaces of N dimensions. For
the Riemann space EN, where N > 2, in which the Ricci tensor results from the
substitution of expression (5.6.17) in expression (5.6.16), thus
Rij ¼ R
N
gij ð5:6:17Þ
where the ratio RN defines a scalar. The space in which the Ricci tensor is pro-
portional to the metric tensor is called the Einstein space.
The scalar curvature of the Einstein space is given by
gpiRij ¼ K
N
gpigij
following for the Ricci tensor with variance (1, 1)
Rpj ¼
K
N
δ pi
The covariant derivative of this expression with respect to variable xp is given by
∂pR
p
j ¼
K
N
∂δpj
∂xj
¼ 0
and with expression (5.4.11)
∂pR
p
j ¼
1
2
∂R
∂xj
¼ 0
whereby
∂R
∂xj
¼ 0 ð5:6:18Þ
then the Einstein space has constant curvature, i.e., is isotropic.
The multiplying of expression (5.6.18) by vector uj allows researching the
eigenvalues of the Ricci tensor, thus
5.6 Particular Cases of Riemann Spaces 275
Riju
j ¼ R
N
giju
j ¼ R
N
ui ) Rij � R
N
δij
� �
uj ¼ 0
where the scalar curvature is constant then the eigenvalues are equal to RN. In this
case the eigenvectors of tensor Rij are undetermined.
Exercise 5.5 Calculate the components of the curvature tensor Rijk‘, of the Ricci
tensor Rij, the scalar curvature and the Gauß curvature K for the bidimensional
spherical space which metric is given by
ds2 ¼ r2 dφ2 þ sin 2φdθ2� �
The metric tensor, the determinant g, and the conjugated tensor of gij are given,
respectively, by
gij ¼
r2 0
0 r2 sin 2φ
" #
g ¼ r4 sin 2φ gij ¼
1
r2
0
0
1
r2 sin 2φ
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For the partial derivatives of the metric tensor the result is g11, 1 ¼ g22, 2 ¼ 0,
following for the Christoffel symbols
Γ111 ¼ Γ222 ¼ Γ211 ¼ Γ212 ¼ Γ221 ¼ 0
Γ112 ¼ Γ121 ¼
g11g11, 2
2
¼ 1
2r2 sin 2φ
∂ r2 sin 2φð Þ
∂φ
¼ � cos φ
sin φ
Γ122 ¼ �
g11g22, 1
2
¼ 1
2r2
∂ r2 sin 2φð Þ
∂φ
¼ � sin φ � cos φ
thus
Rijk‘ ¼
∂Γ ij‘
∂xk
� ∂Γ
i
jk
∂x‘
þ Γmj‘Γ imk � ΓmjkΓ im‘
R1212 ¼ g1mRm212