Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.14 Non-Cartesian coordinates

The other integral theorems discussed in chapter 11 can be extended in a

similar way. For example, written in tensor notation Stokes’ theorem states that,

for a vector fieldai,
∫

S

ijk

∂ak

∂xj

nˆidS=

∮

C

akdxk.

For a general tensor field this has the straightforward extension
∫

S

ijk

∂Tlm···k···n

∂xj

nˆidS=

∮

C

Tlm···k···ndxk.

26.14 Non-Cartesian coordinates

So far we have restricted our attention to the study of tensors when they are

described in terms of Cartesian coordinates and the axes of coordinates are rigidly

rotated, sometimes together with an inversion of axes through the origin. In the

remainder of this chapter we shall extend the concepts discussed in the previous

sections by considering arbitrary coordinate transformations from one general

coordinate system to another. Although this generalisation brings with it several

complications, we shall find that many of the properties of Cartesian tensors

are still valid for more general tensors. Before considering general coordinate

transformations, however, we begin by reminding ourselves of some properties of

general curvilinear coordinates, as discussed in chapter 10.

The position of an arbitrary pointPin space may be expressed in terms of the

three curvilinear coordinatesu 1 ,u 2 ,u 3. We saw in chapter 10 that ifr(u 1 ,u 2 ,u 3 )is

the position vector of the pointPthen atPthere exist two sets of basis vectors

ei=

∂r

∂ui

and i=∇ui, (26.52)

wherei=1, 2 ,3. In general, the vectors in each set neither are of unit length nor

form an orthogonal basis. However, the setseiandiare reciprocal systems of

vectors and so

ei·j=δij. (26.53)

In the context of general tensor analysis, it is more usual to denote the second

setofvectorsiin (26.52) byei, the index being placed as a superscript to

distinguish it from the (different) vectorei, which is a member of the first set in

(26.52). Although this positioning of the index may seem odd (not least because

of the possibility of confusion with powers) it forms part of a slight modification

to the summation convention that we will adopt for the remainder of this chapter.

This is as follows: any lower-case alphabetic index that appears exactly twice in

any term of an expression,once as a subscript and once as a superscript,istobe

summed over all the values that an index in that position can take (unless the