Spring 2006 - Eggers' Class - Exam 2 (Version 1)

Spring 2006 - Eggers' Class - Exam 2 (Version 1) - Math 20E...

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Unformatted text preview: Math 20E Midterm Exam 2 Solution Version 1 1. (6 points) The vector field F ( x, y, z ) = ( y 2 z 3 , 2 xyz 3 , 3 xy 2 z 2 ) is the gradient of the function f ( x, y, z ) = xy 2 z 3 . Let c be the path between (0 , , 0) and (1 , 3 , 1) defined for ≤ t ≤ 1 by c ( t ) = ( t 2 , 3 t cos (2 πt ) , 1 ln 2 t ln(1 + t ) ) . Evaluate Z c F · d s , Since F = ∇ f , R c F · d s = R c ∇ f · d s = f (1 , 3 , 1)- f (0 , , 0) = 9. 2. (6 points) Use an appropriate change of variables to evaluate Z Z R ( x + y ) e x 2- y 2 dx dy, where R is the rectangle bounded by the lines x- y = 0, x- y = 2, x + y = 0, and x + y = 6. Let u = x + y and v = x- y . Then x = 1 2 ( u + v ), y = 1 2 ( u- v ), and ∂ ( x,y ) ∂ ( u,v ) = det 1 2 1 2- 1 2 1 2 = 1 2 . Then, Z Z R ( x + y ) e x 2- y 2 dx dy = Z 6 u =0 Z 2 v =0 ue uv 1 2 dv du = 1 2 Z 6 u =0 u 1 u e uv 2 v =0 du = 1 2 Z 6 ( e 2 u- 1 ) du = 1 4 ( e 12- 13 ) 3. Let S be the rectangle with vertices (0 , 1 , 0), (1 , 1 , 1), (2 , 1 , 0), and (1 , 1 ,...
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Spring 2006 - Eggers' Class - Exam 2 (Version 1) - Math 20E...

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