quiz10' - i . Please clearly label both answers. curl F = i...

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Quiz 10 Solutions 1. Let D = { ( x, y ) R 2 | x 2 + y 2 1 } and f ( x, y ) = 3 x 2 + y 2 + p x 2 + y 2 + y 2 x 2 + y 2 . Compute R R D f ( x, y ) dA . Let P ( x, y ) = - y p x 2 + y 2 and Q ( x, y ) = x ( x 2 + y 2 ). Then ∂P ∂y = - ± p x 2 + y 2 + y 2 x 2 + y 2 ² and ∂Q ∂x = 3 x 2 + y 2 . So, f ( x, y ) = ∂Q ∂x - ∂P ∂y . By Green’s Theorem, if C is the counterclockwise unit circle, R R D f ( x, y ) dA = R C P dx + Q dy . C can be parametrized by x ( t ) = cos t , y ( t ) = sin t for 0 t 2 π . So, Z C P dx + Q dy = Z C x dy - y dx = Z 2 π 0 ± x dy dt - y dx dt ² dt = Z 2 π 0 ( cos 2 t + sin 2 t ) dt = 2 π. 2. Find the curl and divergence of the vector field F ( x, y, z ) = h xe - y , xz - e y + e - y , ze y
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Unformatted text preview: i . Please clearly label both answers. curl F = i j k x y z xe-y xz-e y + e-y ze y = ( ze y-x ) i-(0-0) j + ( z + xe-y ) k = ( ze y-x ) i + ( z + xe-y ) k . div F = x ( xe-y ) + y ( xz-e y + e-y ) + z ( ze y ) = e-y-e y-e-y + e y = 0 . 1...
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