UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering,Department of Civil Engineering Mechanics and MaterialsFall 2003 Professor: S. GovindjeeHigher Order DeterminantsThe determinant of a second order tensor can be expressed in several ways.Suppose T is a second order tensor in n-dimensions, then for a given coor-dinate systemdet[T ] = T1i1T2i2· · · Tninei1i2···in, (1)where i1, i2, · · · , inare indices that range over the set {1, 2, 3, · · · , n}. Thevalue of the permutation symbol1is +1 if (i1, i2, · · · , in) form an even per-mutation and −1 if they form and odd permutation; otherwise, it is zero.Other expressions for the determinant aredet[T ]ej1j2···jn= Tj1i1Tj2i2· · · Tjninei1i2···in(2)anddet[T ] =1n!Tj1i1Tj2i2· · · Tjninei1i2···inej1j2···jn, (3)where j1, j2, · · · jnare indices that range over the set {1, 2, 3, · · · , n}. Inorder to use the expressions above one needs a definition of an even andodd permutation. To determine whether or not a sequence (i1, i2, · · · , in) isan even permutation or an odd permutation, one begins with the sequence(1, 2, 3, · · · , n) and counts the number of interchanges of numbers in thesequence required to generate (i1, i2, · · · , in). If the number of interchangesis an even number then the permutation is even if the number is odd thenthe permutation is odd. As an example the permutation (i1, i2, · · · , in) =(1, 4, 5, 3, 2) can be created by the following series of interchanges(1, 2, 3, 4, 5) → (1, 5, 3, 4, 2) → (1, 5, 4, 3, 2) → (1, 4, 5, 3, 2) . (4)Thus there are 3 interchanges and the permutation is considered odd. Thisimplies that e14532= −1. All these formulae readily reduce to the expressionsgiven in class for the case of n = 3.1Other names for the permutation symbol are alternating symbol and Levi-Civita sym-bol. Some texts consider the permutation symbol to be the components of a tensor andothers do not; the issue is the precise definition of a tensor used in different texts. Forour purposes the idea of the permutation symbol will be sufficient and we will not needto consider its tensorial properties or lack thereof .1Advanced references:1. D. Lovelock and H. Rund, Tensors, Differential Forms, and VariationalPrinciples, Dover, 1989.2. W.L. Burke, Applied Differential Geometry, Cambridge Press,