Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

Let us begin, however, by noting that we may use the Levi–Civita symbol to

write an expression for the determinant of a 3×3 matrixA,

|A|lmn=AliAmjAnkijk, (26.29)

which may be shown to be equivalent to the Laplace expansion (see chapter 8).§

Indeed many of the properties of determinants discussed in chapter 8 can be

proved very efficiently using this expression (see exercise 26.9).

Evaluate the determinant of the matrix

A=





21 − 3

34 0

1 − 21



.

Settingl=1,m=2andn= 3 in (26.29) we find

|A|=ijkA 1 iA 2 jA 3 k

= (2)(4)(1)−(2)(0)(−2)−(1)(3)(1) + (−3)(3)(−2)

+ (1)(0)(1)−(−3)(4)(1) = 35,

which may be verified using the Laplace expansion method.

We can now show that theijkare in fact the components of a third-order tensor.

Using (26.29) with the general matrixAreplaced by the specific transformation

matrixL, we can rewrite the RHS of (26.28) in terms of|L|

′lmn=LliLmjLnkijk=|L|lmn.

SinceLis orthogonal its determinant has the value unity, and so′lmn=lmn.

Thus we see that′lmnhas exactly the properties ofijkbut withi, j, kreplaced by

l, m, n, i.e. it is the same as the expressionijkwritten using the new coordinates.

This shows thatijkis a third-order Cartesian tensor.

In addition to providing a convenient notation for the determinant of a matrix,

δijandijkcan be used to write many of the familiar expressions of vector

algebra and calculus as contracted tensors. For example, provided we are using

right-handed Cartesian coordinates, the vector producta=b×chasasits

ith componentai=ijkbjck; this should be contrasted with the outer product

T=b⊗c, which is a second-order tensor having the componentsTij=bicj.

§This may be readily extended to anN×NmatrixA,i.e.

|A|i 1 i 2 ···iN=Ai 1 j 1 Ai 2 j 2 ···AiNjNj 1 j 2 ···jN,

wherei 1 i 2 ···iNequals 1 ifi 1 i 2 ···iNis an even permutation of 1, 2 ,.. .,Nand equals−1ifitisan

odd permutation; otherwise it equals zero.