Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
which shows that @A=@x are not the components of a tensor because of the
second term on the right hand side. The same also applies to the di€erential of
a contravariant vector. But we can construct a tensor by the following device.
From Eq. (1.111) we have
ˆ @x @x @ x
@ x @ x @x ‡
@2 x @ x
@ x @ x
@x :
Multiplying (1.114) by A and subtracting from (1.113), we obtain
@ A
@ x A
ˆ @A
@x
@x A
@x
@ x @ x :
…1:114†
…1:115†
If we de®ne
then (1.115) can be rewritten as
A ; ˆ @A
@x A ;
…1:116†
A ; ˆ A ;
@x
@ x @x
@ x ;
which shows that A ; is a covariant tensor of rank 2. This tensor is called the
covariant derivative of A with respect to x . The semicolon denotes covariant
di€erentiation. In a Cartesian coordinate system, the Christo€el symbols vanish,
and so covariant di€erentiation reduces to ordinary di€erentiation.
The contravariant derivative is found by raising the index which denotes di€erentiation:
A ; ˆ g A ;:
…1:117†
We can similarly determine the covariant derivative of a tensor of arbitrary
rank. In doing so we ®nd the following simple rule helps greatly:
Thus,
To obtain the covariant derivative of the tensor T with respect to
x , we add to the ordinary derivative @T =@x for each covariant
index …T:† a term T:, and for each contravariant index
† a term ‡ T
…T
... .
T ; ˆ @T
@x T T ;
T ; ˆ @T
@x T ‡ T :
The covariant derivatives of both the metric tensor and the Kronnecker delta
are identically zero (Problem 1.38).
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