Tensores2
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Tensores2


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r
r rrg r= =v v \ufffd e N
r
g r\u3b8\u3b8 \u3b8\u3b8 \u3b8= =v v \ufffd 
\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 (62) \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 
a) Componentes contravariantes: 
2
2
i j k
i id x dx dxa
jk dt dtdt
\u23a7 \u23ab= + \u23a8 \u23ac\u23a9 \u23ad
; ou seja: 
 
N N N N
2
2
2
0 0 0
r
r
r r r rd r dr dr dr d d dr d da r r
rr r rdt dt dt dt dt dt dt dtdt
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8\u3b8 \u3b8 \u3b8\u3b8
\u2212
\u23a7 \u23ab \u23a7 \u23ab \u23a7 \u23ab \u23a7 \u23ab= + + + + = \u2212\u23a8 \u23ac \u23a8 \u23ac \u23a8 \u23ac \u23a8 \u23ac\u23a9 \u23ad \u23a9 \u23ad \u23a9 \u23ad \u23a9 \u23ad
\ufffd\ufffd\ufffd 
 
N N N N
2
2
0 1/ 1/ 0
2
r r
d dr dr dr d d dr d d ra
rr r rdt dt dt dt dt dt dt dt rdt
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8\u3b8\u3b8 \u3b8 \u3b8\u3b8
\u23a7 \u23ab \u23a7 \u23ab \u23a7 \u23ab \u23a7 \u23ab= + + + + = +\u23a8 \u23ac \u23a8 \u23ac \u23a8 \u23ac \u23a8 \u23ac\u23a9 \u23ad \u23a9 \u23ad \u23a9 \u23ad \u23a9 \u23ad
\ufffd\ufffd\ufffd\ufffd 
 
b) Componentes físicos [use a Eq. (7-7) e o Prob. 52]: 
 
N
2
1
r
r rra g a r r\u3b8= = \u2212 \ufffd\ufffd\ufffd e N 2
r
a g a r r\u3b8\u3b8 \u3b8\u3b8 \u3b8 \u3b8= = +\ufffd\ufffd \ufffd\ufffd 
 INTRODUÇÃO AOS TENSORES 63 
\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 (64) \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 
Devemos calcular ;
ji
i j
V
V x
t
\u3b4
\u3b4 = \ufffd , isto é 
; ;
; ;
/
/
r r r r
r
V t V r V
V t V r V
\u3b8
\u3b8 \u3b8 \u3b8 \u3b8
\u3b4 \u3b4 \u3b8
\u3b4 \u3b4 \u3b8
\u23a7 = +\u23aa\u23a8 = +\u23aa\u23a9
\ufffd\ufffd
\ufffd\ufffd 
 
Cálculo dos componentes covariantes ji ji
z
V Z
x
\u2202\u239b \u239e=\u239c \u239f\u239d \u23a0\u2202 
N N
2
0
cos sen ( sen cos )sen sen sen cosr x y x y
y x
x yV V V V V r r r r
r r
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u2212
\u2202 \u2202= + = + = \u2212 = \u2212\u2202 \u2202 
N N
2 2 2
0
sen cos ( sen cos ) cos sen cos cosx y x y
y x
x yV V V V r V r r r r r r\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8\u3b8 \u3b8 \u2212
\u2202 \u2202= + =\u2212 + = \u2212 = \u2212\u2202 \u2202
 
Parametrização da reta 1 (a abscissa será o parâmetro : )y x x t t x= + = 
2 2 2 2 2( 1) ( ) 2 2 1
11 ( ) arctanarctan / arctan
x t
r x y x x r t t t
tx ty x
tx
\u3b8\u3b8
=
\u23a7 \u23a7= + = + + = + +\u23aa \u23aa\u23af\u23af\u23af\u2192\u23a8 \u23a8 ++\u23aa \u23aa == = \u23a9\u23a9
 
 
 Dessa parametrização obtemos as seguintes expressões, necessárias mais adiante: 
2 1
( )
dr tr
dt r t
+= =\ufffd , 2
1
( )r t
\u3b8 \u2212=\ufffd , 1sen ( )
( )
tt
r t
\u3b8 += , cos ( )
( )
tt
r t
\u3b8 = 
 
Cálculo das derivadas covariantes ;
i
i j kj
kV
V V
ijx
\u2202 \u23a7 \u23ab= \u2212 \u23a8 \u23ac\u2202 \u23a9 \u23ad
 
 Adotando a notação explicada no Prob. 58, temos 
 
N N
2
;
0 0
sen sen cosrr r r
rVV V V
rr rrr \u3b8
\u3b8 \u3b8 \u3b8 \u3b8\u2202 \u23a7 \u23ab \u23a7 \u23ab= \u2212 \u2212 = \u2212\u23a8 \u23ac \u23a8 \u23ac\u2202 \u23a9 \u23ad \u23a9 \u23ad 
 
 
N N
;
0 1/
22 sen cos cos
r
r r
r
rVV V V
r r
r r
\u3b8 \u3b8
\u3b8
\u3b8 \u3b8\u3b8
\u3b8 \u3b8 \u3b8
\u2202 \u23a7 \u23ab \u23a7 \u23ab= \u2212 \u2212\u23a8 \u23ac \u23a8 \u23ac\u2202 \u23a9 \u23ad \u23a9 \u23ad
= \u2212 2 2sen sen cos cosr r r\u3b8 \u3b8 \u3b8 \u3b8+ \u2212 +
2sen cos senr r\u3b8 \u3b8 \u3b8= +
 
 
 
N N
;
0 1/
2 2
2
2 sen cos 2 cos sen cos cos
sen cos cos
r r
r
rV
V V V
r rr
r r r r
r r
\u3b8\u3b8 \u3b8
\u3b8
\u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8
\u2202 \u23a7 \u23ab \u23a7 \u23ab= \u2212 \u2212\u23a8 \u23ac \u23a8 \u23ac\u2202 \u23a9 \u23ad \u23a9 \u23ad
= \u2212 \u2212 +
= \u2212
 
 INTRODUÇÃO AOS TENSORES 64 
 
 
N N
;
0
2 2 2 2cos sen
r
r
rV
V V V
r r
\u3b8\u3b8 \u3b8 \u3b8
\u3b8
\u3b8\u3b8 \u3b8\u3b8\u3b8
\u3b8 \u3b8
\u2212
\u2202 \u23a7 \u23ab \u23a7 \u23ab= \u2212 \u2212\u23a8 \u23ac \u23a8 \u23ac\u2202 \u23a9 \u23ad \u23a9 \u23ad
= \u2212 2 2 22 sen cos senr r\u3b8 \u3b8 \u3b8+ + 2
2 2 2
sen cos
cos sen cos
r
r r
\u3b8 \u3b8
\u3b8 \u3b8 \u3b8
\u2212
= +
 
 
Cálculo das derivadas intrínsecas 
 
; ;
2 2
2
( 1) / ( ) / ( )
2 1 1
sen ( ) sen ( )cos ( ) ( )sen ( ) cos ( ) ( )sen ( )( ) ( )
sen ( ) 2 sen ( ) 2( 1)cos ( ) 0
( )
r
r r r
t r t t r t
V
V r V
t
t
t t t r t t t r t tr t r t
t t t t t
r t
\u3b8
\u3b4 \u3b8\u3b4
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8
+
= +
+ \u2212\u23a1 \u23a4 \u23a1 \u23a4= +\u2212 +\u23a3 \u23a6 \u23a3 \u23a6
\u23a7 \u23ab= \u2212 + =\u23a8 \u23ac\u23a9 \u23ad
\ufffd\ufffd
\ufffd\ufffd\ufffd	\ufffd\ufffd
 \ufffd\ufffd\ufffd	\ufffd\ufffd
 
 
 [ ] [ ]
[ ]
; ;
2 2 2 2
2
0
2 1 1
sen cos cos cos sen cos( ) ( )
cos 02 sen 2( 1)cos
r
V
V r V
t
t
r r r rr t r t
t t
\u3b8 \u3b8 \u3b8 \u3b8
\u3b4 \u3b8\u3b4
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
\u3b8\u3b8 \u3b8
= +
+ \u2212= +\u2212 +
= =\u2212 +
\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd	\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
 
 
 Observe que, sobre a reta dada, ( , ) (0,1)x yV V = , um campo eqüipolente; por isso que a 
derivada intrínseca é nula ao longo daquela reta. 
 
\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 (65) \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 
a) Em coordenadas polares 
Componentes covariantes: 
r
\u3c6\u2202
\u2202 e 
\u3c6
\u3b8
\u2202
\u2202 
Componentes físicos: ( ) 1grad r
rh r r
\u3c6 \u3c6\u3c6 \u2202 \u2202= =\u2202 \u2202 e ( )
1 1grad
h r\u3b8 \u3b8
\u3c6 \u3c6\u3c6 \u3b8 \u3b8
\u2202 \u2202= =\u2202 \u2202 
 
b) Em coordenadas esféricas 
Componentes covariantes: 
r
\u3c6\u2202
\u2202 , 
\u3c6
\u3b8
\u2202
\u2202 e 
\u3c6
\u3d5
\u2202
\u2202 
Componentes físicos: 
( ) 1grad r
rh r r
\u3c6 \u3c6\u3c6 \u2202 \u2202= =\u2202 \u2202 , ( )
1 1grad
h r\u3b8 \u3b8
\u3c6 \u3c6\u3c6 \u3b8 \u3b8
\u2202 \u2202= =\u2202 \u2202 e ( )
1 1grad
senh r\u3d5 \u3d5
\u3c6 \u3c6\u3c6 \u3d5 \u3b8 \u3d5
\u2202 \u2202= =\u2202 \u2202 
 
 
 INTRODUÇÃO AOS TENSORES 65 
\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 (66) \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 
/r r r rF F h F= = , / /F F h F r\u3b8 \u3b8 \u3b8 \u3b8= = , / / ( senF F h F r\u3d5 \u3d5 \u3d5 \u3b8 \u3b8= = ) , 2 seng r \u3b8= 
 
( )
( ) ( )
( ) ( )
2 2 2
2
2
2
1div
1
sen sen sen
sen sen
1 1 1
sen
sen sen
i i
i
r
r
F g F
g x
FFF r r rrr r r
F
Fr F
r r rr
\u3d5\u3b8
\u3d5
\u3b8
\u3b8 \u3b8 \u3b8\u3b8 \u3d5\u3b8 \u3b8
\u3b8\u3b8 \u3b8 \u3b8 \u3d5
\u2202= \u2202
\u23a1 \u23a4\u2202 \u2202 \u2202 \u239b \u239e= + + \u239c \u239f\u23a2 \u23a5\u239d \u23a0\u2202 \u2202 \u2202\u23a3 \u23a6
\u2202\u2202 \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 (67) \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 
 Neste problema adotamos a notação explicada no Prob. 58. 
 ( ) ( )2 ; ;
; ;
( , ) rr rrrr
r
r g g g g
r
\u3b8\u3b8 \u3b8\u3b8\u3b8\u3b8 \u3b8
\u3c6 \u3c6\u3c6 \u3b8 \u3c6 \u3c6 \u3b8
\u2202 \u2202\u2207 = + = +\u2202 \u2202 
 Usando o Prob. 52, temos que 
 1 1rrrrg g= \u21d2 = 
 2 21/g r g r\u3b8\u3b8\u3b8\u3b8 = \u21d2 = 
 
 Abaixo usamos os símbolos de Christoffel em coordenadas polares já calculados no Prob. 
56: 
 
( ) ( )
N N
2
2
;
0 0
r
r
rr rrr rr r r
\u3b8\u3c6 \u3c6 \u3c6\u3c6 \u3c6
\u3b8
\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202\u2202 \u2202= \u2212 \u2212 =\u23a8 \u23ac \u23a8 \u23ac\u2202 \u2202 \u2202\u2202 \u2202 \u2202\u23a9 \u23ad \u23a9 \u23ad 
( ) ( )
N N
2
2
;
0r
r
r
r r\u3b8
\u3b8\u3c6 \u3c6 \u3c6 \u3c6\u3c6 \u3c6
\u3b8\u3b8 \u3b8\u3b8\u3b8 \u3b8\u3b8 \u3b8 \u3b8
\u2212
\u23a7 \u23ab \u23a7 \u23ab\u2202 \u2202 \u2202 \u2202 \u2202\u2202 \u2202= \u2212 \u2212 = +\u23a8 \u23ac \u23a8 \u23ac\u2202 \u2202 \u2202 \u2202\u2202 \u2202 \u2202\u23a9 \u23ad \u23a9 \u23ad 
 
( )2 2 222 2 2 2 2 221 1 1( , ) 1r r r rrr r r r\u3c6 \u3c6 \u3c6 \u3c6\u3c6 \u3c6\u3c6 \u3b8 \u3b8\u3b8\u2202 \u2202 \u2202 \u2202\u2202 \u2202\u2207 = \u22c5 + = + ++ \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 (68) \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 
33
2
33
3 1
32 2
g
g x
\u2202\u23a7 \u23ab =\u23a8 \u23ac \u2202\u23a9 \u23ad , i.e., 
1
2
g
g
\u3d5\u3d5
\u3d5\u3d5
\u3d5
\u3d5\u3b8 \u3b8
\u2202\u23a7 \u23ab =\u23a8 \u23ac \u2202\u23a9 \u23ad . 
 
\u2022 Cálculo do 1o membro: 
 
 
2 2 2x y z
x y z
\u3d5 \u3d5 \u3d5 \u3d5
\u3d5\u3b8 \u3d5 \u3b8 \u3d5 \u3b8 \u3d5 \u3b8
\u23a7 \u23ab \u2202 \u2202 \u2202 \u2202 \u2202 \u2202= + +\u23a8 \u23ac \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202 \u2202\u23a9 \u23ad
 
 
 INTRODUÇÃO AOS TENSORES 66 
 Tendo em conta que 
 
 sen cosx r \u3b8 \u3d5= , sen seny r \u3b8 \u3d5= , cosz r \u3b8= 
 
temos que 
2
cos senx r \u3b8 \u3d5\u3d5 \u3b8
\u2202 = \u2212\u2202 \u2202 , 
2
cos cosy r \u3b8 \u3d5\u3d5 \u3b8
\u2202 =\u2202 \u2202 
e que 
2
2 2 2 2 2
2 2 2 2 2
/ sen sen sen
sen1 ( / ) sen
1/ sen cos cosarctan
sen1 ( / ) sen
0
y x y r
x ry x x y r
y x x r
x y ry x x y r
z
\u3d5 \u3b8 \u3d5 \u3d5
\u3b8\u3b8
\u3d5 \u3b8 \u3d5 \u3d5\u3d5 \u3b8\u3b8
\u3d5
\u23a7 \u2202 \u2212 \u2212 \u2212 \u2212= = = =\u23aa \u2202 + +\u23aa\u23aa \u2202= \u21d2 = = = =\u23a8 \u2202 + +\u23aa\u23aa \u2202\u23aa =\u2202\u23a9
 
 Logo, 
 
( ) 2 2cossen( cos sen ) ( cos cos ) cot sen cot cos cotsensenr r rr\u3d5 \u3d5\u3d5\u3b8 \u3d5 \u3b8 \u3d5 \u3b8 \u3d5 \u3b8 \u3d5 \u3b8\u3d5\u3b8 \u3b8\u3b8\u23a7 \u23ab \u2212= \u2212 + = + =\u23a8 \u23ac\u23a9 \u23ad \u25a0
 
 
\u2022 Cálculo do 2o membro: 
2 2 2 2
2 2
1 1sen 2 sen cos cot
2 2 sen
g
g h r r
g r
\u3d5\u3d5
\u3d5\u3d5 \u3d5
\u3d5\u3d5
\u3b8 \u3b8 \u3b8 \u3b8\u3b8 \u3b8
\u2202= = \u21d2 = =\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 (69) \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 1 ij
ji g g xxg
\u3c6\u3c6 \u2202\u2202 \u239b \u239e\u2207 = \u239c \u239f\u2202\u2202 \u239d \u23a0 
 
11 22 33
1 2 31 2 3
1 1 1g g g g g g
x x xx x xg g g
\u3c6 \u3c6 \u3c6\u2202 \u2202 \u2202\u2202 \u2202 \u2202\u239b \u239e \u239b \u239e \u239b \u239e= + +\u239c \u239f \u239c \u239f \u239c \u239f\u239d \u23a0 \u239d \u23a0 \u239d \u23a0\u2202 \u2202 \u2202\u2202 \u2202 \u2202 
 
2
1
senr \u3b8=
2 senr
r
\u3b8\u2202\u2202 ( ) 22 11 sen rr r\u3c6 \u3b8\u3b8 \u2202\u2202 +\u22c5 \u22c5 \u2202\u2202 21sen r\u3b8
2
1
senr
\u3c6
\u3b8
\u3b8
\u2202\u239b \u239e\u239c \u239f\u2202\u239d \u23a0
+ 2 senr \u3b8\u3d5
\u2202
\u2202 2 2
1
senr
\u3c6
\u3d5\u3b8
\u2202\u239b \u239e\u239c \u239f\u2202\u239d \u23a0
 
 
( ) ( ) 222 2 2 2 21 1 1sensen senrr rr r r \u3c6\u3c6 \u3c6\u3b8\u3b8 \u3b8\u3b8 \u3b8 \u3d5\u2202 \u2202 \u2202\u2202 \u2202= + +\u2202 \u2202\u2202 \u2202 \u2202 \u25a0 
 
 INTRODUÇÃO AOS TENSORES 67 
\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 (70) \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( )ug h h h u u\u3d5= = +v v v 11 2 2 2
1 1 1uu
uu u
g g
g h u
= = = = +v 
 
 22 2 2 2
1 1 1g g
g h u
= = = = +
vv
vv v v
 33 2 2 2
1 1 1g g
g h u
\u3d5\u3d5
\u3d5\u3d5 \u3d5
= = = =
v
 
 
 
11 22 332
1 2 31 2 3
1 g g g g g g
g x x xx x x
\u3c8 \u3c8 \u3c8\u3c8 \u23a7 \u23ab\u2202 \u2202 \u2202\u2202 \u2202 \u2202\u23a1 \u23a4 \u23a1 \u23a4 \u23a1 \u23a4\u2207 = + +\u23a8 \u23ac\u23a2 \u23a5 \u23a2 \u23a5 \u23a2 \u23a5\u2202 \u2202 \u2202\u2202 \u2202 \u2202\u23a3 \u23a6 \u23a3 \u23a6 \u23a3 \u23a6\u23a9 \u23ad 
 
2 2
2 2
1 ( )