Mathematical Methods for Physicists: A concise introduction - Site Map

COVARIANT DIFFERENTIATION
where the functions
( )
ˆ ˆ g ‰; Š
…1:111†
are the Christo€el symbol of the second kind.
Eq. (1.110) is, of course, a set of N coupled di€erential equations; they are the
equations of the geodesic. In Euclidean spaces, geodesics are straight lines. In a
Euclidean space, g are independent of the coordinates x , so that the Christo€el
symbols identically vanish, and Eq. (1.110) reduces to
with the solution
d 2 x
ds 2 ˆ 0
x ˆ a s ‡ b ;
where a and b are constants independent of s. This solution is clearly a straight
line.
The Christo€el symbols are not tensors. Using the de®ning Eqs. (1.109) and the
transformation of the metric tensor, we can ®nd the transformation laws of the
Christo€el symbol. We now give the result, without the mathematical details:
@x
@x @x
; ˆ ;
@ x @ x @ x ‡ g
@x @ 2 x
@x @ x @ x :
…1:112†
The Christo€el symbols are not tensors because of the presence of the second term
on the right hand side.
Covariant di€erentiation
We have seen that a covariant vector is transformed according to the formula
A ˆ @x
@ x A ;
where the coecients are functions of the coordinates, and so vectors at di€erent
points transform di€erently. Because of this fact, dA is not a vector, since it is the
di€erence of vectors located at two (in®nitesimally separated) points. We can
verify this directly:
@ A
@ x ˆ @A @x @x
@x @ x @ x ‡ A @ 2 x
@ x @ x ;
…1:113†
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