Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.18 Derivatives of basis vectors and Christoffel symbols

the outer product of the two tensors, or any contraction of them, is a relative

tensor of weightw 1 +w 2. As a special case, we may useijkandijkto construct

pseudovectors from antisymmetric tensors and vice versa, in an analogous way

to that discussed in section 26.11.

For example, if theAijare the contravariant components of an antisymmetric

tensor (w=0)then

pi=^12 ijkAjk

are the covariant components of a pseudovector (w=−1), sinceijkhas weight

w=−1. Similarly, we may show that

Aij=ijkpk.

26.18 Derivatives of basis vectors and Christoffel symbols

In Cartesian coordinates, the basis vectorseiare constant and so their derivatives

with respect to the coordinates vanish. In a general coordinate system, however,

the basis vectorseiandeiare functions of the coordinates. Therefore, in order

that we may differentiate general tensors we must consider the derivatives of the

basis vectors.

First consider the derivative∂ei/∂uj. Since this is itself a vector, it can be

written as a linear combination of the basis vectorsek,k=1, 2 ,3. If we introduce

the symbol Γkijto denote the coefficients in this combination, we have

∂ei

∂uj

=Γkijek. (26.75)

The coefficient Γkijis thekth component of the vector∂ei/∂uj.Usingthereci-

procity relationei·ej=δij, these 27 numbers are given (at each point in space)

by

Γkij=ek·

∂ei

∂uj

. (26.76)

Furthermore, by differentiating the reciprocity relationei·ej=δijwith respect

to the coordinates, and using (26.76), it is straightforward to show that the

derivatives of the contravariant basis vectors are given by

∂ei

∂uj

=−Γikjek. (26.77)

The symbol Γkijis called aChristoffel symbol(of the second kind), but, despite

appearances to the contrary, these quantities donotform the components of a

third-order tensor. It is clear from (26.76) that in Cartesian coordinates Γkij=0

for all values of the indicesi,jandk.