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IndexNotationHints - b j e j = a i b j e i × e j = a i b j...

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ETH Zurich Department of Mechanical and Process Engineering Institute for Mechanical Systems Center of Mechanics Prof. Dr. Edoardo Mazza 1 Useful rules for index notation The permutation ε ijk is defined as follows, ε ijk = 1 (i,j,k)=(123),(231),(312) - 1 (i,j,k)=(132),(213),(321) 0 (all other combinations) . (1) Such that when the indices are in increasing order the value is postive 1, and when the indices are in decreasing order the value is negative 1. Otherwise it is 0. This definition is motivated by the need for defining an easy way to write the cross product. e i × e j = ε ijk e k . (2) Given this notation, we can easily express the cross product between vectors a and b as, a × b = ( a i e i ) × ( b j
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Unformatted text preview: ( b j e j ) = a i b j e i × e j = a i b j ε ijk e k (3) Remarks: The Kronecker delta can be used as identity matrix: I ˆ= δ ij (4) The permutation has the properties that, ε ijk = ε jki = ε kij (5) and, ε ijk =-ε jik . (6) There is exists the following relation between the permutation and the Kronecker delta, 1. ε ijk ε pqk = δ ip δ jq-δ iq δ jp (7) Here i,j,p,q are free indices and must match on both sides of the equation. k is a dummy index. 2. ε ijk ε pjk = 2 δ ip (8) Here i,p are free indices and must match on both sides of the equation. j,k are dummy indices. 3. ε ijk ε ijk = 6 (9) 1...
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