# Berkeley CIVENG C231 - Higher Order Determinants (2 pages)

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## Higher Order Determinants

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## Higher Order Determinants

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University of California, Berkeley
Course:
Civeng C231 - Mechanics of Solids
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UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering Department of Civil Engineering Mechanics and Materials Fall 2003 Professor S Govindjee Higher Order Determinants The 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 coordinate system det T T1i1 T2i2 Tnin ei1 i2 in 1 where i1 i2 in are indices that range over the set 1 2 3 n The value of the permutation symbol1 is 1 if i1 i2 in form an even permutation and 1 if they form and odd permutation otherwise it is zero Other expressions for the determinant are det T ej1 j2 jn Tj1 i1 Tj2 i2 Tjn in ei1 i2 in 2 and 1 Tj i Tj i Tjn in ei1 i2 in ej1 j2 jn 3 n 1 1 2 2 where j1 j2 jn are indices that range over the set 1 2 3 n In order to use the expressions above one needs a definition of an even and odd permutation To determine whether or not a sequence i1 i2 in is an even permutation or an odd permutation one begins with the sequence 1 2 3 n and counts the number of interchanges of numbers in the sequence required to generate i1 i2 in If the number of interchanges is an even number then the permutation is even if the number is odd then the 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 det T 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 This implies that e14532 1 All these formulae readily reduce to the expressions given in class for the case of n 3 1 Other names for the permutation symbol are alternating symbol and Levi Civita symbol Some texts consider the permutation symbol to be the components of a tensor and others do not the issue is the precise definition of a tensor used in different texts For our purposes the idea of the permutation symbol will be sufficient and we will not need to consider its tensorial properties or lack thereof 1 Advanced references 1 D Lovelock and H Rund Tensors Differential Forms and Variational Principles Dover 1989 2 W L Burke Applied Differential Geometry Cambridge Press 1985 2

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