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Cartesian tensors
Published in D.E. Bourne, P.C. Kendall, Vector Analysis and Cartesian Tensors, 2017
Let i, j, k be a permutation of the numbers 1, 2, 3. The permutation is said to be even if i, j, k are three consecutive numbers of the set 1, 2, 3, 1, 2, and is said to be odd otherwise. For example, 2, 3, 1 is an even permutation of the numbers 1, 2, 3, and 2, 1, 3 is an odd permutation. It may be readily verified that all even permutations of the numbers 1, 2, 3 can be brought about by an even number of interchanges of pairs of these numbers and all odd permutations by an odd number of interchanges.
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
(The determinant (call it D) on the right is known as the Vander Monde determinant. If we set, for i < j, ai = aj in D, then two columns become identical, and hence D = 0. This means that (ai − aj) is a factor of D. Further the coefficient of a2a32…ann−1 on both sides is 1. Hence the last equality.) The even permutations in Sn form a subgroup An of Sn, called the alternating group of degree n.Proof. If σ1 and σ2 are even permutations in Sn, then each of them is a product of an even number of transpositions and hence so is their product. Further if σ=t1t2…tr, where each ti is a transposition, then σ−1=tr−1tr−1−1…t2−1t1−1. Hence the inverse of an even permutation is even. Further the identity permutation is even since for any transposition t, e = t° t = t2.A permutation is odd or even according to whether it is expressible as a product of an odd number or even number of transpositions.
Fast accurate seakeeping predictions
Published in Ship Technology Research, 2020
Equations (46) and (47) for translational and rotational motion are combined to one matrix equation for the six accelerations. To that end, the vector product involving in (46) is written in index notation: for arbitrary three-dimensional vectors and their vector product can be written (using Einstein summation) aswhere are the elements of the Levi–Civita tensor: is 1 if ijk constitute an even permutation of (1,2,3); for an odd permutation; and 0 if i,j,k are not all different. Using this notation, (46) and (47) can be combined into the matrix equationHere E is the 3 by 3 unit matrix, 0 the 3 by 3 zero matrix, and A the 6 by 6 added mass matrix. The rigid mass matrix on the left-hand side (in parentheses) may appear unusual. The reason is: here moments and moments of inertia refer to the centre of gravity G, but the translation refers to the origin of the body-fixed coordinate system, which usually differs from G. If both coincide, we have , causing the upper right 3 by 3 submatrix to vanish.