e2s - Math 481 Exam 2 (SOLUTIONS), Friday, April 6, 2007 1....

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Math 481 Exam 2 (SOLUTIONS), Friday, April 6, 2007 1. Let W be a (0 , 2)-tensor on a 2-manifold M such that in a chart ( U, φ = ( x 1 , x 2 )) we have W 1 , 1 = x 1 + 5 , W i,j = 0 for ( i, j ) 6 = (1 , 1) . Let ( V, ψ = ( y 1 , y 2 )) be another chart such that V = U and such that y 1 = x 1 + 3 x 2 , y 2 = - x 1 . (a) Find all the coefficients W 0 i,j of W in the chart ( V, ψ ), as explicit functions of y 1 , y 2 . (b) In the chart ( U, φ ) compute the function W ( ( x 2 ) 2 dx 1 , 3 dx 1 + e x 1 x 2 dx 2 ) . Solution. (a) We have W 0 i,j = X k,` ∂y i ∂x k ∂y j ∂x ` W k,` = ∂y i ∂x 1 ∂y j ∂x 1 W 1 , 1 = ( x 1 + 5) ∂y i ∂x 1 ∂y j ∂x 1 . Note that y 1 = x 1 + 3 x 2 , y 2 = - x 1 implies x 1 = - y 2 , x 2 = 1 3 ( y 1 + y 2 ). Hence W 0 1 , 1 = ( x 1 + 5) · 1 · 1 = x 1 + 5 = - y 2 + 5 W 0 1 , 2 = ( x 1 + 5) · 1 · - 1 = y 2 - 5 W 0 2 , 1 = ( x 1 + 5) · - 1 · 1 = y 2 - 5 W 0 2 , 2 = ( x 1 + 5) · - 1 · - 1 = - y 2 + 5 . (b) We have
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e2s - Math 481 Exam 2 (SOLUTIONS), Friday, April 6, 2007 1....

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