math192_hw11sp06 - HW11 Solutions 16.3 Path Independence,...

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Unformatted text preview: HW11 Solutions 16.3 Path Independence, Potential Functions, and Conservative Fields 2 32. P y = 0 = N z . M z = 0 = P x , M y = 2 x sin y = N x F isconservative thereexistsanfsothat F = f ; f x = 2 x cos y f ( x, y, z ) = x 2 cos y + g ( y, z ) f y = x 2 sin y + g y = x 2 sin y g y = g ( y, z ) = h ( z ) f ( x, y, z = x 2 cos y + h ( z ) f z = h ( z ) = 0 h ( z ) = C f ( x, y, z ) = x 2 cos y + C F = ( x 2 cos y ) (a) integraltext C 2 x cos ydx x 2 sin ydy = [ x 2 cos y ] (0 , 1) (1 , 0) = 0 1 = 1 (b) integraltext C 2 x cos ydx x 2 sin ydy = [ x 2 cos y ] (1 , 0) (- 1 , ) = 1 ( 1) = 2 (c) integraltext C 2 x cos ydx x 2 sin ydy = [ x 2 cos y ] (1 , 0) (- 1 , 0) = 1 1 = 0 (d) integraltext C 2 x cos ydx x 2 sin ydy = [ x 2 cos y ] (1 , 0) (1 , 0) = 1 1 = 0 33....
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math192_hw11sp06 - HW11 Solutions 16.3 Path Independence,...

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