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HW1Solution - AE 321 – Solution to Homework Problems(Fall...

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Unformatted text preview: AE 321 – Solution to Homework Problems (Fall ’08) Chapter 1: Mathematical Preliminaries 1.1 ik jk ij C B A Let’s first work with the dummy index and perform the summation ik k i k i k i C B A B A B A 3 3 2 2 1 1 (1.2) In expression (1.2) we have 2 free indices, therefore it represents 9 equations that are obtained by expanding (1.2) for every combination of i and k : i.e. 1 i & 1 k , 1 i & 2 k , …. The 9 equations are given in the following table. 1 k 2 k 3 k 1 i 11 31 13 21 12 11 11 C B A B A B A 12 32 13 22 12 12 11 C B A B A B A 13 33 13 23 12 13 11 C B A B A B A 2 i 21 31 23 21 22 11 21 C B A B A B A 22 32 23 22 22 12 21 C B A B A B A 23 33 23 23 22 13 21 C B A B A B A 3 i 31 31 33 21 32 11 31 C B A B A B A 32 32 33 22 32 12 31 C B A B A B A 33 33 33 23 32 13 31 C B A B A B A 1.2 k k i i C B A , Once again, let’s start by working with the dummy index, i.e. k , and perform the summation A i B i ,1 C 1 B i ,2 C 2 B i ,3 C 3 B i x 1 C 1 B i x 2 C 2 B i x 3 C 3 (1.3) In this case we only have one free index; therefore expression (1.3) represents 3 equations: i 1 A 1 B 1,1 C 1 B 1, 2 C 2 B 1,3 C 3 B 1 x 1 C 1 B 1 x 2 C 2 B 1 x 3 C 3 i 2 A 2 B 2,1 C 1 B 2,2 C 2 B 2,3 C 3 B 2 x 1 C 1 B 2 x 2 C 2 B 2 x 3 C 3 i 3 A 3 B 3,1 C 1 B 3,2 C 2 B 3, 3 C 3 B 3 x 1 C 1 B 3 x 2 C 2 B 3 x 3 C 3 2.1 Notice that the first expression survives only if...
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HW1Solution - AE 321 – Solution to Homework Problems(Fall...

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