Mathematical Methods for Physics and Engineering - Matematica.NET

26.14 NON-CARTESIAN COORDINATESThe other integral theorems discussed in chapter 11 can be extended in asimilar way. For example, written in tensor notation Stokes’ theorem states that,for a vector field a i ,∫∮∂a kɛ ijk ˆn i dS = a k dx k .S ∂x j CFor a general tensor field this has the straightforward extension∫Sɛ ijk∂T lm···k···n∂x jˆn i dS =∮CT lm···k···n dx k .26.14 Non-Cartesian coordinatesSo far we have restricted our attention to the study of tensors when they aredescribed in terms of Cartesian coordinates and the axes of coordinates are rigidlyrotated, sometimes together with an inversion of axes through the origin. In theremainder of this chapter we shall extend the concepts discussed in the previoussections by considering arbitrary coordinate transformations from one generalcoordinate system to another. Although this generalisation brings with it severalcomplications, we shall find that many of the properties of Cartesian tensorsare still valid for more general tensors. Before considering general coordinatetransformations, however, we begin by reminding ourselves of some properties ofgeneral curvilinear coordinates, as discussed in chapter 10.The position of an arbitrary point P in space may be expressed in terms of thethree curvilinear coordinates u 1 ,u 2 ,u 3 . We saw in chapter 10 that if r(u 1 ,u 2 ,u 3 )isthe position vector of the point P then at P there exist two sets of basis vectorse i = ∂r and ɛ i = ∇u i , (26.52)∂u iwhere i =1, 2, 3. In general, the vectors in each set neither are of unit length norform an orthogonal basis. However, the sets e i and ɛ i are reciprocal systems ofvectors and soe i · ɛ j = δ ij . (26.53)In the context of general tensor analysis, it is more usual to denote the secondset of vectors ɛ i in (26.52) by e i , the index being placed as a superscript todistinguish it from the (different) vector e i , which is a member of the first set in(26.52). Although this positioning of the index may seem odd (not least becauseof the possibility of confusion with powers) it forms part of a slight modificationto 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 inany term of an expression, once as a subscript and once as a superscript, istobesummed over all the values that an index in that position can take (unless the955