M402.set.3 - j system, using the appropriate matrix in the...

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Math 402 Problem Set III 1. A coordinate system, v i , is defined as follows: v 1 = 2y -x -z v 2 = -x +y +z v 3 = 3z -y a) Find the contravariant transformation matrix, A . b) Find the covariant transformation matrix, B . c) Find the metric tensor for the v i system. d) Find the inverse of the metric tensor for the v i system. e) Find v^ i in terms of the x^ j . f) Find the u k vectors in the v i system in terms of v i and v^ j . g) Find the u n vectors in the v i system in terms of v i and v^ j . 2. In the v i system of problem 1 a "vector", G , is defined as follows: G = v 2 u 1 + 5 v 3 u 2 - v 1 u 3 . a) Find the contravariant components, G' k , of G in the v i system . b) Transform the G' k found in part a) to their "counterparts" in the x
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Unformatted text preview: j system, using the appropriate matrix in the following equation: G i = A i n G' n (leave results in terms of the v j variables). c) Find G·x^ , G·y^ , and G·z^ from your answer to part b). Hint: this is easy! d) Find the covariant components, G' j , in the v i system . e) Evaluate the following in the v i system, leaving answer in terms of the v j : ( δ j m δ k n- δ j n δ k m ) G' j G' m u n ·u k . 3. Evaluate the following in the v i system of problem 1: a) u 1 ·u 3 b) u 3 ·u 1 c) u 2 ·u 3...
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This note was uploaded on 12/20/2010 for the course PHYS 402 taught by Professor J. during the Fall '09 term at Missouri S&T.

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