math192_hw11sp06

# 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 ,π ) = 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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