Tensores2
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Tensores2


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1
2
ii
i
ii
i g
ii g x
\u2202\u23a7 \u23ab =\u23a8 \u23ac \u2202\u23a9 \u23ad (8-12) 
 1
2
ii
j
ii
i g
ij g x
\u2202\u23a7 \u23ab =\u23a8 \u23ac \u2202\u23a9 \u23ad
 onde , 0 e com
a convenção da soma suspensa
ij i j
i j g \u2260\u2260 = (8-13) 
 1
2
jj
i
ii
gi
jj g x
\u2202\u23a7 \u23ab = \u2212\u23a8 \u23ac \u2202\u23a9 \u23ad
 (8-14) 
 
 Na literatura, em vez de { }kij , também se usam { , }ij k e kij\u393 ; esta última notação, entretan-
to, sugere um caráter tensorial que, como veremos adiante, não é verdadeiro em geral. 
 
 
9. Lei de transformação dos símbolos de Christoffel 
Equation Section (Next) 
 Considere o símbolo de Christoffel de 1a espécie no sistema de coordenadas ix\u2032 : 
 1[ , ] ;
2
jk ijik
i j k
g gg
ij k
x x x
\u2032 \u2032\u2202 \u2202\u2032\u2202\u239b \u239e\u2032 \u2261 + \u2212\u239c \u239f\u239d \u23a0\u2202 \u2202 \u2202\u2032 \u2032 \u2032 (9-1) 
para obtê-lo no sistema de coordenadas ix , calculemos nesse sistema o primeiro termo entre 
parênteses, fazendo uso da regra da cadeia e da lei de transformação da métrica: 
 INTRODUÇÃO AOS TENSORES 20 
( ) ( )2 2m n l m n m n m njk mnmn mni i j k l i j k i j k j i kg gx x x x x x x x xg gx x x x x x x x x x x x x x\u2032\u2202 \u2202\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202= = + +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 . 
 
Desta equação, com duas permutações ( / / / )i j j ki j kjk ik ijg x g x g x\u2032 \u2032 \u2032\u2202 \u2202 \u23af\u23af\u23af\u2192\u2202 \u2202 \u23af\u23af\u23af\u2192\u2202 \u2202\u2032 \u2032 \u2032R R , ob-
temos no sistema ix os dois últimos termos da Eq. (9-1); substituindo nesta os resultados, en-
contramos 
 
 
2 21 1[ , ]
2 2
l m n m n m n
mn
mnl i j k i j k j i k
g x x x x x x xij k g
x x x x x x x x x x
\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 = + +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032
2 21 1
2 2
l m n m n m n
mn
mnl j i k j i k i j k
l m
g x x x x x x xg
x x x x x x x x x x
\u239b \u239e\u239c \u239f\u239d \u23a0
\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202+ + +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032
R
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
2
, , , ,
1 1
2 2
l m n m n
mn
mnl k i j k i j
l m n n l m
g x x x x xg
x x x x x x x
\u2192
\u239b \u239e\u239c \u239f\u239d \u23a0
\u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2212 \u2212\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
2m n
i k j
x x
x x x
\u2202 \u2202+ \u2202 \u2202 \u2202\u2032 \u2032 \u2032
\u239b \u239e\u239c \u239f\u239d \u23a0
 
 
21 1 1
2 2 2
l m n m l n n l m m n
mn ln lm
mnl i j k m j i k n k i j i j k
g g gx x x x x x x x x x xg
x x x x x x x x x x x x x x x
\u2202 \u2202 \u2202\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202= + \u2212 +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 
21
2
l m n m n
mn ln lm
mni j k l m n i j k
g g gx x x x xg
x x x x x x x x x
\u2202 \u2202 \u2202\u2202 \u2202 \u2202 \u2202 \u2202\u239b \u239e= + \u2212 +\u239c \u239f\u239d \u23a0\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 
 
(onde indicamos as trocas de índices de acordo com o rodapé da p. 8), ou 
 
 
2
[ , ] [ , ] .
l m n m n
mni j k i j k
x x x x xij k lm n g
x x x x x x
\u2202 \u2202 \u2202 \u2202 \u2202\u2032 = +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 (9-2) 
 
Esta é a lei de transformação do símbolo de Christoffel de 1a espécie. Observe que o segundo 
termo no 2o membro impede que [ij,k] se transforme como um tensor do tipo 03 (covariante de 3
a 
ordem). 
 Calculemos agora o símbolo de Christoffel de 2a espécie no sistema de coordenadas ix\u2032 
em termos desses símbolos no sistema de coordenadas ix . Usando a Eq. (9-2), temos 
 
 
2
2
[ , ] [ , ]
[ , ]
n
b
n
b
k s l m n m n
ks ab
mna b i j s i j s
k l m s n k m s n
ab
a i j b s a i j b s
k x x x x x x xg ij s g lm n g
ij x x x x x x x x
x x x x x x x x xg lm n g
x x x x x x x x x x
\u3b4
\u3b4
\u2032\u23a7 \u23ab \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u239b \u239e\u2032 \u2032= = +\u23a8 \u23ac \u239c \u239f\u239d \u23a0\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032\u23a9 \u23ad
\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032= +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd
{ }
2
[ , ]
a
m
ab
mn
k l m k m
an
a i j m i j
a
lm
g
x x x x xg lm n
x x x x x x
\u3b4
\u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032= +\u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd
 
 INTRODUÇÃO AOS TENSORES 21 
 
ou, trocando a por n, 
 
 
2
.
k l m k m
n i j m i j
k nx x x x x
ij lmx x x x x x
\u2032\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032= +\u23a8 \u23ac \u23a8 \u23ac\u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032\u23a9 \u23ad \u23a9 \u23ad (9-3) 
 
Esta é a lei de transformação do símbolo de Christoffel de 2a espécie. Novamente note que é o 
segundo termo no 2o membro que impede que { }kij se transforme como um tensor do tipo 12 . 
 Da (9-3) podemos calcular em termos dos símbolos de Christoffel de 2a espécie uma ex-
pressão para 2 /m i jx x x\u2202 \u2202 \u2202\u2032 \u2032 . Multiplicando tal equação por /a kx x\u2202 \u2202 \u2032 , obtemos 
 
 
2
,
a a
n m
a a k l m a k m
k k n i j k m i j
k nx x x x x x x x
ij lmx x x x x x x x x
\u3b4 \u3b4
\u2032\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032= +\u23a8 \u23ac \u23a8 \u23ac\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032\u23a9 \u23ad \u23a9 \u23ad\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd
 \ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd
 
donde 
 
2
.
a a l m
i j k i j
k ax x x x
ij lmx x x x x
\u2032\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202= \u2212\u23a8 \u23ac \u23a8 \u23ac\u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032\u23a9 \u23ad \u23a9 \u23ad (9-4) 
 
Nesta expressão podemos inverter x e x\u2032 para obter 
 
 
2
.
a a l m
i j k i j
k ax x x x
ij lmx x x x x
\u2032\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032= \u2212\u23a8 \u23ac \u23a8 \u23ac\u2202 \u2202 \u2202 \u2202 \u2202\u23a9 \u23ad \u23a9 \u23ad (9-5) 
 
 
10. Derivada covariante 
Equation Section (Next) 
 a) Derivada covariante de tensores 
 Exceto no caso da diferenciação de uma função ( )ix\u3c6 invariante, para a qual / jx\u3c6\u2202 \u2202 \u2032 = 
( / ) ( / )i i jx x x\u3c6\u2202 \u2202 \u2202 \u2202 \u2032 , mostrando que / ix\u3c6\u2202 \u2202 é um vetor covariante, as derivadas parciais de 
tensores não resultam em novos tensores. Considere, por exemplo, um vetor contravariante aV \u2032 ; 
diferenciando em relação a nx\u2032 ambos os membros de sua lei de transformação, 
 
 ,
a
a j
j
xV V
x
\u2202 \u2032\u2032 = \u2202 
obtemos 
 
 
2
.
a i a a i j i a
j j
n n i j j n i n i j
V x x x x V x xV V
x x x x x x x x x x
\u2032\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032\u239b \u239e= = +\u239c \u239f\u239d \u23a0\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 
 
No 2o membro, o 2o termo impede que /a nV x\u2032\u2202 \u2202 \u2032 se transforme como um tensor de 2a ordem 
do tipo 11 . Entretanto, eliminando a derivada segunda 
2 /a i jx x x\u2202 \u2202 \u2202\u2032 que aparece naquele termo 
por meio da Eq. (9-5), encontramos 
 INTRODUÇÃO AOS TENSORES 22 
 
 
m l
n
a a i j i a l m
j
n j n i n k i j
j k
a i k i a m i l
j j
k n i n k j n i
V
k aV x x V x x x xV
ij lmx x x x x x x x
k ax x V x x x x xV V
ijx x x x x x x x
\u3b4
\u2192
\u2032
\u2032\u23a1 \u23a4\u2032 \u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032= + \u2212\u23a8 \u23ac \u23a8 \u23ac\u23a2 \u23a5\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u23a9 \u23ad \u23a9 \u23ad\u23a3 \u23a6
\u23a7 \u23ab\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032= + \u2212\u23a8 \u23ac\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032\u23a9 \u23ad
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd
 \ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd
,
a i k
j m
k n i
lm
k ax x V V V
ij nmx x x
\u2032\u23a7 \u23ab\u23a8 \u23ac\u23a9 \u23ad
\u2032\u23a1 \u23a4\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202\u2032 \u2032= + \u2212\u23a8 \u23ac \u23a8 \u23ac\u23a2 \u23a5\u2202 \u2202 \u2202\u2032 \u23a9 \u23ad \u23a9 \u23ad\u23a3 \u23a6
 
 
ou 
 
 ,
a a i k
m j
n k n i
a kV x x VV V
nm ijx x x x
\u2032 \u23a1 \u23a4\u23a1 \u23a4\u2032 \u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202\u2032\u2032+ = +\u23a8 \u23ac \u23a8 \u23ac\u23a2 \u23a5\u23a2 \u23a5\u2202 \u2202 \u2202 \u2202\u2032 \u2032\u23a9 \u23ad \u23a9 \u23ad\u23a3 \u23a6 \u23a3 \u23a6
 (10-1) 
 
onde vemos que os termos entre colchetes é um tensor de 2a ordem do tipo 11 (pois se transforma 
como tal); a expressão desses termos é usada para definir a derivada covariante do vetor contra-
variante kV (em relação a ix e com respeito à métrica ijg incorporada nos símbolos de Chris-
toffel) e é denotada de várias maneiras: 
 
 ; ou ou ou ;
k k
j k k k
i i ii i
kV DVV V V V
ijx Dx
\u23a7 \u23ab\u2202 + \u2261 \u2207\u23a8 \u23ac\u2202 \u23a9 \u23ad &
 (10-2) 
 
aqui daremos preferência às duas primeiras formas. 
 A Eq. (10-2) mostra que à derivada parcial /k iV x\u2202 \u2202 devemos adicionar um termo "corre-
tivo", { }j kijV no caso, para obtermos um tensor: a derivada covariante ;k iV . Veremos que isso 
vale para qualquer tensor ijT
"
" : ; /
i i n
j n jT T x= \u2202 \u2202" "" " + termos "corretivos". 
 Se considerarmos agora um vetor covariante jV \u2032 e diferenciarmos em relação a ix\u2032 a sua 
lei de transformação, /a jj aV V x x\u2032 = \u2202 \u2202 \u2032 , obtemos 
 
 
2
.
a a k a
j a
a ai i j j i k i j
V Vx x x xV V
x x x x x x x x
\u2032\u2202 \u2202\u2202 \u2202 \u2202 \u2202 \u2202\u239b \u239e= = +\u239c \u239f\u239d \u23a0\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032 \u2032 
 
Nesta, eliminando a derivada segunda por meio da Eq. (9-4), encontramos 
 
,
a k a l m
j a
ai j i k k i j
V k aVx x x x xV
ij lmx x x x x x x
\u2032\u2032\u2202 \u23a1 \u23a4\u2202 \u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202 \u2202= + \u2212\u23a8 \u23ac \u23a8 \u23ac\u23a2 \u23a5\u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u2032 \u2032 \u2032 \u2032 \u2032 \u2032\u23a9 \u23ad \u23a9 \u23ad\u23a3 \u23a6
 
 
ou 
 
 INTRODUÇÃO AOS TENSORES 23 
( )
 e 
k
a
j j
a ki k i
V
a k l m
a
aj i k i j
a m k l a k
m l l m
m
j i l i j
V Vk kxV V
ij ijx x x
aVx x x x V
lmx