Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
A generalization to V N is immediate. We de®ne the line element ds in V N to be
given by the following quadratic form, called the metric form, or metric
ds 2 ˆ X3
X 3
ˆ1 ˆ1
g dx dx ˆ g dx dx :
…1:100†
For the special cases of rectangular coordinates and spherical coordinates, we
have
0 1
0
1
1 0 0
1 0 0
B C
B
~g ˆ…g †ˆ@
0 1 0A; ~g ˆ…g †ˆ 0 r 2 C
@ 0 A: …1:101†
0 0 1
0 0 r 2 sin 2
In an N-dimensional orthogonal coordinate system g ˆ 0 for 6ˆ . And in a
Cartesian coordinate system g ˆ 1andg ˆ 0 for 6ˆ . In the general case of
Riemannian space, the g are functions of the coordinates x … ˆ 1; 2; ...; N†.
Since the inner product of g and the contravariant tensor dx dx is a scalar
(ds 2 , the square of line element), then according to the quotient law g is a
covariant tensor. This can be demonstrated directly:
Now dx 0 ˆ…@x 0 =@x †dx ; so that
ds 2 ˆ g dx dx ˆ g 0 dx 0 dx 0 :
or
g 0 @x 0 @x 0
@x @x dx dx ˆ g dx dx
!
g 0 @x 0 @x 0
@x @x g dx dx ˆ 0:
The above equation is identically zero for arbitrary dx , so we have
g ˆ
0 @x @x 0
@x @x g ;
0
…1:102†
which shows that g is a covariant tensor of rank two. It is called the metric
tensor or the fundamental tensor.
Now contravariant and covariant tensors can be converted into each other with
the help of the metric tensor. For example, we can get the covariant vector (tensor
of rank one) A from the contravariant vector A :
A ˆ g A :
…1:103†
Since we expect that the determinant of g does not vanish, the above equations
can be solved for A in terms of the A . Let the result be
A ˆ g A :
…1:104†
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