Tensores2
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Tensores2


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normal a S, denotado por in , é 
2| |
i
i ij ij
j i j
Nn g g
x x xN
\u3c6 \u3c6 \u3c6\u2202 \u2202 \u2202= = \u2202 \u2202 \u2202 \u25a0 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (49) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
 Usando as definições de divergência e derivada covariante, podemos escrever 
 ;div
i
i i j
i i
iFF F F
ijx
\u23a7 \u23ab\u2202= = + \u23a8 \u23ac\u2202 \u23a9 \u23ad
 
 Mas, usando o Prob. 37c, temos que 
 1 1 / 1
2 2
j
j j
i gg g x
ij g g g gx x
\u2202\u23a7 \u23ab \u2202 \u2202 \u2202= = =\u23a8 \u23ac \u2202 \u2202\u23a9 \u23ad ; 
logo, 
 ( )1 1div i j ii i ii j i i ig gF F FF g F g Fg g gx x x x x\u239b \u239e\u2202 \u2202\u2202 \u2202 \u2202= + = + =\u239c \u239f\u239c \u239f\u2202 \u2202 \u2202 \u2202 \u2202\u239d \u23a0 \u25a0 
 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (51) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
( ) ( ) ( ) ( ); ; ; ; ,i i ir ir s ir s ir s i sjk kj r jk r kj rjk s srjk rsjk sjkV V g V V g R V g R V g R V R V\u2212 = \u2212 = = = \u2212 = \u2212
 
onde usamos os Probs. 26 e 50a. 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (52) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
1o modo: 
 ijg é tirado direto da expressão do quadrado do elemento de comprimento de arco. Este 
modo convém quando o sistema de coordenadas for ortogonal e com fatores de escala conheci-
dos, em cujo caso 2 2 1 2 2 2 21 2( ) ( )ds h dx h dx= + +", donde 211 1g h= , 222 2g h= , etc e 
0ij i jg \u2260 = . Além disso, sendo ( )
ijg a inversa da matriz diagonal ( )ijg , temos que 
11
111/g g=
, 22 221/g g= , etc, pois a matriz inversa 1( )ija\u2212 de uma matriz diagonal 1 2( ) diag ( , , )ija \u3b1 \u3b1= " 
é a matriz diagonal 1( )ija
\u2212 = 1 2diag (1/ , 1/ , )\u3b1 \u3b1 " . 
 INTRODUÇÃO AOS TENSORES 56 
 
Coordenadas polares: 
2 2 2 2 2
2 2
1 0 1 0
, ,
0 0 1/
rr r
rr r
r
r
g g g g
ds dr r d g r
g g r rg g
\u3b8\u3b8
\u3b8 \u3b8\u3b8\u3b8 \u3b8\u3b8
\u3b8 \u239b \u239e\u239b \u239e \u239b \u239e\u239b \u239e= + \u21d2 = = =\u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f \u239c \u239f \u239c \u239f\u239c \u239f\u239d \u23a0 \u239d \u23a0 \u239d \u23a0\u239d \u23a0
 
 
Coordenadas cilíndricas: 
2 2 2 2 2ds d d dz\u3c1 \u3c1 \u3d5= + + \u21d2 
2 2
2
1 0 01 0 0
10 0 , 0 0 ,
0 0 1
0 0 1
z
z
z
z
z z zz
z z zz
g g gg g g
g g g g g g g
g g g g g g
\u3c1\u3c1 \u3c1\u3d5 \u3c1
\u3c1\u3c1 \u3c1\u3d5 \u3c1
\u3d5\u3c1 \u3d5\u3d5 \u3d5\u3d5\u3c1 \u3d5\u3d5 \u3d5
\u3c1 \u3d5\u3c1 \u3d5
\u3c1 \u3c1\u3c1
\u239b \u239e\u239b \u239e\u239b \u239e\u239b \u239e \u239c \u239f\u239c \u239f\u239c \u239f\u239c \u239f \u239c \u239f\u21d2 = = =\u239c \u239f\u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f\u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f\u239d \u23a0 \u239c \u239f\u239d \u23a0 \u239d \u23a0 \u239d \u23a0
 
 
Coordenadas esféricas: 
2 2 2 2 2 2 2sends dr r d r d\u3b8 \u3b8 \u3d5= + + \u21d2 
2 2 4 2
2 2 2 2 1
1 0 0 1 0 0
0 0 , 0 0 , sen
0 0 sen 0 0 ( sen )
rr r r
rr r r
r
r
r
r
g g gg g g
g g g r g g g r g r
g g g r g g g r
\u3b8 \u3d5
\u3b8 \u3d5
\u3b8 \u3b8\u3b8 \u3b8\u3d5\u3b8 \u3b8\u3b8 \u3b8\u3d5
\u3d5 \u3d5\u3b8 \u3d5\u3d5\u3d5 \u3d5\u3b8 \u3d5\u3d5
\u3b8
\u3b8 \u3b8
\u2212
\u2212
\u239b \u239e\u239b \u239e \u239b \u239e\u239b \u239e \u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f = = =\u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f \u239c \u239f\u239c \u239f \u239c \u239f\u239d \u23a0 \u239d \u23a0\u239d \u23a0 \u239d \u23a0
2o modo: 
 São usadas as fórmulas 
 e
i j
ijk k
ij i j
k k
z z x xg g
z zx x
\u2202 \u2202 \u2202 \u2202= = \u2202 \u2202\u2202 \u2202 
onde iz são coordenadas cartesianas e 
ix são coordenadas curvilíneas. Assim, nas coordenadas 
polares, temos: 
2 2cos sen 1rr
x x y yg
r r r r
\u3b8 \u3b8\u2202 \u2202 \u2202 \u2202= + = + =\u2202 \u2202 \u2202 \u2202 
2 2 2 2 2sen cosx x y yg r r r\u3b8\u3b8 \u3b8 \u3b8\u3b8 \u3b8 \u3b8 \u3b8
\u2202 \u2202 \u2202 \u2202= + = + =\u2202 \u2202 \u2202 \u2202 
sen cos sen cos 0r r
x x y yg r r g
r r\u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8\u3b8 \u3b8
\u2202 \u2202 \u2202 \u2202= + = \u2212 + = =\u2202 \u2202 \u2202 \u2202 
 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (53) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
Componentes contravariantes: 
 Estas são calculadas através da equação 
i
i j
j
xV V
x
\u2202 \u2032\u2032 = \u2202 , com 
1,2
1,2
, coordenadas e os componentes dados
no sistema de coordenadas cartesianas,
jj
jj
x y
x x y
V V V
=
=
\u23af\u23af\u23af\u2192
\u23af\u23af\u23af\u2192
 
 
 INTRODUÇÃO AOS TENSORES 57 
1,2
1,2
, coordenadas e os componentes contrava-
riantes no sistema de coordenadas polares,
ii
ii r
x r
V V V\u3b8
\u3b8=
=
\u23af\u23af\u23af\u2192\u2032
\u2032 \u23af\u23af\u23af\u2192
 
 
 Da lei de transformação de coordenadas dada por cosx r \u3b8= e seny r \u3b8= obtêm-se: 
cos sen
sen cos
x x
rr
y y r
r
\u3b8 \u3b8\u3b8
\u3b8 \u3b8
\u3b8
\u2202 \u2202\u239b \u239e\u239c \u239f \u2212\u239b \u239e\u2202 \u2202 =\u239c \u239f \u239c \u239f\u2202 \u2202\u239c \u239f \u239d \u23a0\u239c \u239f\u2202 \u2202\u239d \u23a0
 e 
cos sen
sen cos
r r
x y
r rx y
\u3b8 \u3b8
\u3b8 \u3b8\u3b8 \u3b8
\u2202 \u2202\u239b \u239e \u239b \u239e\u239c \u239f\u2202 \u2202 \u239c \u239f\u239c \u239f = \u2212\u239c \u239f\u2202 \u2202\u239c \u239f \u239c \u239f\u239d \u23a0\u239c \u239f\u2202 \u2202\u239d \u23a0
 . 
 
 Portanto, 
cos senr x y x y
r rV V V V V
x y
\u3b8 \u3b8\u2202 \u2202= + = +\u2202 \u2202 
sen cos
x y x yV V V V Vx y r r
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8\u2202 \u2202= + = \u2212 +\u2202 \u2202 
 
Componentes covariantes: 
 Estas são calculadas através da equação 
j
i ji
xV V
x
\u2202\u2032 = \u2202 \u2032 , com 
1,2 e , e ,jj j x yx V x y V V
=\u23af\u23af\u23af\u2192 ( coordenadas e componentes no sistema cartesiano ) 
 
1,2 e , e ,ji i rx V r V V\u3b8\u3b8=\u2032 \u23af\u23af\u23af\u2192\u2032 ( coordenadas e componentes covariantes no sistema de 
coordenadas polares ) 
 
 Logo, 
cos senr x y x y
x yV V V V V
r r
\u3b8 \u3b8\u2202 \u2202= + = +\u2202 \u2202 
sen cosx y x y
x yV V V V r V r\u3b8 \u3b8 \u3b8\u3b8 \u3b8
\u2202 \u2202= + = \u2212 +\u2202 \u2202 
 
Componentes físicos [usando a Eq. (7-7), tendo em conta o Prob. 52]: 
 
cos senr r rr r x yV V g V V V\u3b8 \u3b8= = = + 
 
/ sen cosx yV V g V r V V\u3b8 \u3b8 \u3b8\u3b8 \u3b8 \u3b8 \u3b8= = = \u2212 + 
 
 Note que os componentes físicos coincidem com as projeções do vetor nas direções dos 
versores, isto é, r rV V e= \u22c5
G G
 e V V e\u3b8 \u3b8= \u22c5
G G
, onde usamos a notação elementar, na qual 
xV V i= +
G G
 yV j
G
, cos senre i j\u3b8 \u3b8= +
G GG
 e sen cose i j\u3b8 \u3b8 \u3b8= \u2212 +
G GG
. 
 
 
 
 INTRODUÇÃO AOS TENSORES 58 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (54) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
 N N
1 0
r r
r rr rV g V g V V
\u3b8\u3b8= + = ........................ verdade 
 
 N N
2
2
0
r
r
r
V g V g V r V\u3b8 \u3b8\u3b8 \u3b8 \u3b8\u3b8= + = .................... verdade 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (55) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
2 2 2
cos sen (2 )cos 2 sen
(2 cos sen )cos (2 cos sen )sen
2 cos sen cos 2 cos sen
r
x yV V V x y xy
r r r r
r r r
\u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8
= + = \u2212 +
= \u2212 +
= \u2212 +
 
 
2 2
sen cos sen cos(2 ) 2
sen cos(2 cos sen ) (2 cos sen )
2sen cos sen 2 cos sen
x yV V V x y xyr r r r
r r r r
r r
r
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8\u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8
= \u2212 + = \u2212 \u2212 +
= \u2212 \u2212 +
= \u2212 + +
 
 
 Usando o Prob. 54, obtemos 
2 2 22 cos sen cos 2 cos senrrV V r r r\u3b8 \u3b8 \u3b8 \u3b8 \u3b8= = \u2212 + 
2 2 2 2 3 22 sen cos sen 2 cos senV r V r r r\u3b8\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8= = \u2212 + + 
\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 (56) \u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013 
Item (a): Devemos fazer os índices i, j e k tomarem os valores 1 ou 2. 
1[ , ]
2
jk ijik
i j k
g gg
ij k
x x x
\u2202 \u2202\u239b \u239e\u2202= + \u2212\u239c \u239f\u2202 \u2202 \u2202\u239d \u23a0
: 
 
2
1 1[ , ] (1) 0
2 2
1[ , ] 0
2
1[ , ] 0
2
1 1[ , ] ( )
2 2
rr
r rr
rr
g
rr r
r r
g g
rr
r
g
r r
g
r r r
r r
\u3b8
\u3b8\u3b8
\u3b8 \u3b8
\u3b8 \u3b8
\u3b8 \u3b8
\u2202 \u2202= = =\u2202 \u2202
\u2202 \u2202= \u2212 =\u2202 \u2202
\u2202= =\u2202
\u2202 \u2202= = =\u2202 \u2202
 2
[ , ] [ , ] 0
[ , ] [ , ]
1 1[ , ] ( )
2 2
1[ , ] 0
2
r
r r r r
r r r
g g
r r r
r r
g
\u3b8 \u3b8\u3b8
\u3b8\u3b8
\u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8
\u3b8\u3b8 \u3b8
\u3b8\u3b8 \u3b8 \u3b8
= =
= =
\u2202 \u2202 \u2202= \u2212 = \u2212 = \u2212\u2202 \u2202 \u2202
\u2202= =\u2202
 
 
[ , ]ks
k
g ij s
ij
\u23a7 \u23ab =\u23a8 \u23ac\u23a9 \u23ad : 
 INTRODUÇÃO AOS TENSORES 59 
 
NN N
NN N
N N
01 0
01 0
1 0
[ , ] [ , ] 0
[ , ] [ , ] 0
0
[ , ] [ , ]
rr r
rr r
rr r
r
r
g rr r g rr
rr
r
g r r g r
r
r r
r r
r
g r g r
\u3b8
\u3b8
\u3b8
\u3b8
\u3b8 \u3b8 \u3b8\u3b8
\u3b8 \u3b8
\u3b8\u3b8 \u3b8\u3b8 \u3b8\u3b8\u3b8 \u2212
\u23a7 \u23ab = + =\u23a8 \u23ac\u23a9 \u23ad
\u23a7 \u23ab = + =\u23a8 \u23ac\u23a9 \u23ad
\u23a7 \u23ab \u23a7 \u23ab= =\u23a8 \u23ac \u23a8 \u23ac\u23a9 \u23ad \u23a9 \u23ad
\u23a7 \u23ab = + = \u2212\u23a8 \u23ac\u23a9 \u23ad \ufffd\ufffd\ufffd	\ufffd\ufffd
 
NN NN
NN N
N N
2
2
2
0 00 1/
00 1/
00 1/
[ , ] [ , ] 0
1[ , ] [ , ]
1
[ , ] [ , ] 0
r
r
r
rr
r
r
g rr r g rr
rr
g r r g r
r r
r r r
g r g
\u3b8 \u3b8\u3b8
\u3b8 \u3b8\u3b8
\u3b8 \u3b8\u3b8
\u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8\u3b8
\u3b8 \u3b8
\u3b8 \u3b8
\u3b8 \u3b8\u3b8 \u3b8\u3b8 \u3b8\u3b8\u3b8
\u23a7 \u23ab = + =\u23a8 \u23ac\u23a9 \u23ad
\u23a7 \u23ab = + =\u23a8 \u23ac\u23a9 \u23ad
\u23a7 \u23ab \u23a7 \u23ab= =\u23a8 \u23ac \u23a8 \u23ac\u23a9 \u23ad \u23a9 \u23ad
\u23a7 \u23ab = + =\u23a8 \u23ac\u23a9 \u23ad
\ufffd\ufffd\ufffd	\ufffd\ufffd
\ufffd\ufffd\ufffd	\ufffd\ufffd
 
 
Item (b): Pela fórmula 
2
[ , ] s si j k
z z
ij k
x x x
\u2202 \u2202= \u2202 \u2202 \u2202 temos, por exemplo: 
2 2 2 2
2 2
( cos ) ( cos ) ( sen ( sen )[ , ]
( cos ) (cos ) ( sen ) (sen )
x x y y r r r rr
r r r r
r r r
\u3b8 \u3b8 \u3b8 \u3b8\u3b8\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8
\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 ) \u2202= + = + =\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2202 \u2202
= \u2212 + \u2212 = \u2212
 
 
 Pela fórmula 
2 k
s
i j
s
k z x
ij zx x
\u2202\u23a7 \u23ab \u2202=\u23a8 \u23ac \u2202\u2202 \u2202\u23a9 \u23ad
 temos, por exemplo: 
 
2 2 2 2( cos ) (arctan / ) ( sen ) (arctan / )
sen cos 1( sen ) (cos )
x y