Transcript 投影片 1 - National Cheng Kung University

3、general curvilinear coordinates in Euclidean 3-D
3-1 coordinate system and general in Euclidean 3-D
Suppose that general coordinates are( ξ1 ,ξ 2 ,ξ 3 ); this means that the
1
2
position vectors x of a point is a known function of ξ , ξ and ξ3 ,
x
then the choice that is usually made for the base vectors is εi =
i

ξ
For consistency with the right-handedness of the εi , the coordinates
x
x
x
must be numbered in such a way that


0
1
2
3
ξ
ξ
ξ
3
x
 3

x
 2
x
 1
2
1
x
 3
x
 1
x
 2
x
x
x


0
1
2


 3
ξ1  r
As an example , consider the cylindrical coordinate
ξ2  
ξ3  z
i
With X  x eiin terms of the Cartesian coordinates xi and the Cartesian base
1
2
3
vectors e i, where X  x e1  x e2  x e3 , we have
x1  r cos 
x 2  r sin 
x3  z
And so
X [(r cos  )e1  (r sin  )e 2  ze3 ]

 (cos  )e1  (sin  )e2
r
r
X [(r cos  )e1  (r sin  )e2  ze3 ]
ε2 

 (r sin  )e1  ( r cos  )e 2


X [(r cos  )e1  (r sin  )e 2  ze3 ]
ε3 

 e3
z
z
ε1  (cos  )e1  (sin  )e2
1 0 0 
ε 2  (r sin  )e1  (r cos  )e2
g   0 r 2 0 
ij
ε3  e3
 0 0 1 
ε1 
3-2 metric tensor and jacobian
g
We have already seen that ij is a tensor ; it will now be shown why it is
called the metric tensor
x x
x
ε

ε

g
g


The definition i j
ij, together with εi = i ,give ij
i
j

ξ

ξ

ξ
x
dξi
Note that dx 
ξi
So that an element of arc length ds satisfies
x x i j
j
(ds)  dx  dx 

dξ dξ  g dξi dξ
ij
ξi ξ j
2
Note that dx 
x
dξi is the same as
ξi
dx  (dξi )ε
i
The jacobian of the transformation relating Cartesian coordinates and
curvilinear coordinates is determinant of the array xi ξ j and the
element of volume having the vectors
(
x 1 x 2 x 3
dξ ).( 2 dξ ), ( 3 dξ )
1
ξ
ξ
ξ
As edges is
dV  Jdξ1dξ2dξ3  (ε1  ε2  ε3 )dξ1dξ2dξ3
J 
g
3-3 Transformation rule for change of coordinates
i
Suppose a new set general coordinates ξ is introduced, with the
i
understanding that the relations between ξi and  are known, at least in
principle. The rule for changing to new tensor components is
ij
i
j
pq
T ..k  T (ε  ε )(ε  ε )(ε r  ε k )
..r
p
q
ε  g ε j  g ( x
i
ij
ij
ij
ij
ξ
i
),
g

ε

ε
,
and
g
g


j
ij
jk
j
i
k
x x
jt x x
pq is x x
(g

)( g

)( g ru
)
..r
p
s
q
t
u
k
ξ
T ..k  T
ξ
ξ
ξ
ξ
ξ
l
m

x

x

ξ

x

x

ξ
x x ξ n
is
jt
pq
ru
 T (g

)( g

)( g

)
..r
l
s
p
m
t
q
u
n
ξ ξ ξ
ξ ξ ξ k
ξ ξ ξ
i
j
ξ
ξ r
ij
pq ξ
T ..k  T
(
)(
)(
)
..r
p
q
k
ξ
ξ
ξ
4、tensor calculus
4-1 gradient of a scalar
If f  ξ , ξ , ξ
1
2
3

f
is a scalar function, then
f
f


f

But grad
f
x
ξi

f x
j
j
i
x ξ
(
f
x
j
e )
j
x
ξi
j e j ; hence
f
ξi
 (f )  ε
is the i th convariant component of
ξi
f
f 
εi
 i
f
i
f
An alternative way to conclude that
 i is a vector is to note
f
j
d is a scalar for all d  j recall that d  j is a vector ,and
that df 
j

invoke the appropriate quotient law.
4-2 Derivative of a vector ; christoffel` symbol; covariant derivative
Consider the partial derivative
F
ξ
i
j of a vector F. with F = F ε ,
i
we have
F
ξ
ε
write
ξ
j

ε
i
i
ε F
j i
j
F i
ξ
ξ
i  k ε
ij k
j
christoffel system of the second kind
the k th contravriant component of the derivative with respect to  j of the base
vector. Note that
ε
ξ
i 
j
2 x
ξi ξ
ε
j

j
ξi
k   k
ij
ji
We can now write
F
ξ
(
j
F i
ξ
j
 F k i )ε
(4-2-1)
ki i
Introduce the notation
F
ξ
j
 Fi ε
(4-2-2)
,j i
i
F
This means that , j ----- called the convariant derivative of F i --- is definded as
F
th
the i contravariant component of the vector
j comparing (4-2-1) and (4
2-2) then gives us the formula
i

F
Fi 
 F k i
,j
kj
j
ξ
F i
Although
i
ξ
j is not necessarily a tensor,F, j is one , for
dF 
F
ξ
j
j
j
dξ  ( F i dξ )ε
,j
i
th
The covariant derivative of Fi writing as Fi , j, is defined as the i covariant
component of F  j ; hence
Fi , j  gki F, kj
(4-2-3)
A direct calculation of Fi, j is more instructive; with F= F εi ,we have
i
F
ε k
i
i
F ε (
ε F
)
i, j
k j
j
j
ξ
ξ
ξ
F
j
k
k
Now ε  εi  i , whence
k
ε 
i
And therefore
consequently
ε
k
 ε  i  ε k  (l ε )   k
ij l
ij
j
j
ε


ε k

F
i, j

j
  k εi
ij
F

i  F k
k ij
j
F
i
And while this be the same as (4-2-3) it shows the explicit addition to
j

needed to provide the covariant derivative of F
i
Other notations are common for convariant derivatiives; they are, in approximate
order of popularity
F
F
D F  F
i|j i; j
j i
j i
Although ijkis not a third-order tensor, the superscript can nevertheless be
lowered by means of the operation ε
g
 k  [ij , p]  g
kp ij
kp
εk 
ξ
i ε
j
p

ε
ξ
i
j
and the resultant quantity , denoted by [ij, p] , is the Christoffel symbol of the
first kind. The following relations are easily verified
ε
ε
x  x
j
[ij, k ]  ε  i  ε 

k
k
j
ξi
ξ k ξi ξ j
ξ
2
g
ε
ε

ij
j
i ε  ε 

(ε  ε ) 
 [ik , j ]  [ jk , i]
i
j
j
i
k
k
k
k
ξ
ξ
ξ
ξ
g
g

g
1
jk
ij
p
ik
g   [ij, k ]  {


}
kp ij
2  j
 i
 k
[Prove] ：
g

g

g
1
ij
jk
{ ik 

}
2  j
 i  k
1
 {[kj , i ]  [ij, k ]  [ ki, j ]  [ ji, k ]  [ jk , i]  [ik , j ]}
2
1
 {[ij , k ]  [ ji, k ]}
2
 [ij , k ]
(4-2-4)
4-3 covariant derivatives of Nth –order Tensors
ij
Let us work out the formula for the covariant derivatives of A..k. write
ij
ε ε εk
..k i j
A A
By definition
A
ij
 A
ε ε εk
..k , p i j
p
ξ

ij

( A ε ε εk )
..k i j
p
ξ
This leads directly to the formula
ij

..k , p
A
ij
..k  Arj i  Air  j  Aij  r
..k rp
..k rp
..r pk
p
A

4-4 divergence of a vector
A useful formula for
i
p i
i

F
  F  divF  F  (
)

F

i
,i
ip

will be developed for general coordinate systems. We have
i
ip
 g is [ip, s ]
g
g
1 is gis
ps
ip

g [

]
p
i
s
2
ξ
ξ
ξ
1 is gis

g
p
2
ξ
But, by determinant theory
g
is
is
 gg
p
p
g
ξ
( g )
Hence i  1 g  1
ip 2 g
p
g ξ p
ξ
And therefore
1 
 F 
(F i g )
g ξi
ξ
4-5 Riemann-Christoffel Tensor
Since the order of differentiation in repeated partial differentiation of Cartesian
tensor is irrelevant, it follows that the indices in repeated covariant differentiation of
general tensors in Euclidean 3-D may also be interchanged at will. Thus, identities
like
,ij  , ji
and
f k ,ij  f k , ji
(4-5-1)
Eq (4-5-1) is easily verified directly, since
,ij 
 2
ξ i ξ
j
p
, p ij
 
However, the assertion of (4-5-1) in Euclidean 3-D leads to some nontrivial
information. By direct calculation it can be shown that
p
f k ,ij  f k , ji  R.kij
fp
p
p

kj
p
ki   p  r   p  r
R


.kij
ri kj
rj ki
j
ξi
ξ

With help of (4-2-4) it can be shown that R pkij , the Riemann-christoffel tensor,
is given by
2 g
2 g
2 g
2 g
1
pj
pi
kj
ki 
R
 [


]  g [ r  m   r  m ]
pkij 2
rm pj ki
pi kj
ξ k ξi ξ p ξ j ξ k ξ j ξ p ξi
But since the left-hand side of vanishes for all vectors
f k , it follows that
Rpkij  0
(4-5-2)
Although (4-5-2) represents 81equations, most of them are either identities
or redundant, since R pkij   R pkji   Rkpij  Rijpk . Only six distinct
nontrivial conditions are specified by (4-5-2), and they may be written as
Rpkij  R1223  R1231  R2323  R2331  R3131  0
(4-5-3)
[Note]
p
p

kj
ki   p  r   p  r ) f
f
 f
(

k ,ij
k , ji
rj kj
rj ki p
j
ξi

ξ
[prove]
f
k ,ij

f
k ,i
p
p
 f 
kj pi
ij
ξ i
f
f
f
p
p
p

( k  f ) (
  f )   r ( k   f )
ki p
kj
kp r
ij
kp 
ξ i ξ i
ξi
ξ r

f
k , ji


f
(
k p f )p (
kj p
kj
j
ξ i ξ
f
ξ
p
p f
  r f )   ( k   f )
pj r
ji
kp 
j
p
ξ
p
f
f
p
p
p
p
p
p f k
p
k
ki
f


f 

  r f  
   f
k ,ij
ki
kj
kj pi r
ij
ij kp 
j
j p
j
p
ξi
ξ ξ i ξ
ξ
ξ
2 f

p
f
f
k, j
p
p
p
p
p
p
p
k
f




   r f      f
k , ji
kj
ki
kj pj r
ji
ji kr 
j
j
ξi
ξi
ξ i ξ
ξ

2 f
p
p

jk
kj
p
p
f
f
  r f 
f     pjk  r f 
f
k ,ij
k , ji
kj pi r
p
jk
pj r
p
j
i
ξ
ξ

p
p
kj  ki
p
p
(

  r   r ) f
ri kj
rj ki p
j
ξi

ξ
Since Rpkij is antisymmetrical in i and j as well as in p and k, no information is
lost if (4-5-2) is multiplied by ε spk εtij . Consequently , a set of six equations
equivalent to (4-5-3) is given neatly by
S
st
0
Where S st is the symmetrical, second-order tensor
S st 
1 spk tij
ε
ε
R
pkij
4
p
The tensor Sij is related simply to the Ricci tensor R  R
ij
.ijp
p
R S g S
ij
ij
ij p
So that (4-5-3) is also equivalent to the assertion Rij  0
4-6 Integral Relations
The familiar divergence theorem relating integrals over a volume V and its
boundary surface S can obvious be written in tensor notation as
i
i
 f, i dv   f Ni ds
v
s
Where Ni is the unit outward normal vector to S . Similar stokes，theorem
for integrals over a surface S and its boundary line C is just
ijk
k t ds
ε
f
N
ds

f

c
k, j i
k
s
Where tk is the unit tangent vector to C , and the usual handedness rules
apply for direction of Ni and ti
廣義相對論
T    PU U  Pg
1
R  g  R  8GT
2
R  R
ij
ij
Ri
jkl

ij
Rg R
ij


xk
i 
 
 jk 
x
i
 
j 

 i 
 
 k 
 
 
 jk 
i 
i

g
jk ,  

 
 jk 
i 
 
 l 
 
 
 jl 
1  g j  gk  g jk
 jk ,     k  j  
2 x
x
x

ds 2  g dxi dx
ij
j
A
j,kl
A
j,lk
 Ri A
jkl i
R
ijkl
R
klij
Rij kl 
R
 R
jikl
ijkl
R
 R
ijlk
ijkl
i
i
R  R  Ri  0
jkl
klj
ljk
Rijkl  0


2
2
1
nn  1nn  1  1  nn  1n  2n  3  N  n n  1
2
4!
12


xi

A

i
Ai 
 
A

j
,j
j
x
A
i, j




A

i   A
ij 
j