hw1-soln - MAE 261A E NERGY AND VARIATIONAL P RINCIPLES IN...

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MAE 261A E NERGY AND V ARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS F ALL 2004 Homework Assignment No. 1 Solutions Due Wednesday, October 13, 2004. Problem 1 Let u , v , and w be vectors. Using index notation (and the Einstein summation convention), prove the following vector identities: a. ijk pqk = δ ip δ jq - δ iq δ jp Solution: The indicies i , j , and p , q each can assume the values 1,2,3, so ε ijk ε pqk (summed over k ) represents (3 × 3) × (3 × 3) = 9 × 9 = 81 cases. These can be separated into three groups or categories. Group 1. If i = j or p = q , then ε ijk ε pqk vanishes, and so does δ ip δ jq - δ iq δ jp . That leaves (3 × 3 - 3) × (3 × 3 - 3) = 36 cases to consider. Group 2. The situations where i < j and j < i , as well as p < q and q < p , are related: δ ip δ jq - δ iq δ jp satisfies the same relations as ε ijk ε pqk under exchange: ε ijk ε pqk = - ε jik ε pqk , ε ijk ε pqk = - ε ijk ε qpk , ε ijk ε pqk = ε jik ε qpk . That leaves ((3 × 3 - 3) 1 2 ) × ((3 × 3 - 3) 1 2 ) = 9 cases to consider. Group 3. When we look at the remaining cases ( i, j ) = (1 , 2) , (2 , 3) , (3 , 1) , ( p, q ) = (1 , 2) , (2 , 3) , (3 , 1) , we realize that out of those 9, only 6 need to be calculated, since (obviously) ε ijk ε pqk = ε pqk ε ijk . These 6 cases are ε 12 k ε 12 k = 1 (only one contribution, when k = 3 )
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