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Unformatted text preview: xy i + ( x 2 + 2 yz ) j + y 2 k . Is F conservative? If so, compute the most general potential function f of F . Solution: One has ∇ × F = ± ± ± ± ± ± ± ± i j k ∂ ∂x ∂ ∂y ∂ ∂z 2 xy x 2 + 2 yz y 2 ± ± ± ± ± ± ± ± = (2 y2 y ) i(00) j + (2 x2 x ) k = Since, in addition to ∇ × F = , the domain of F is R 3 , which is simplyconnected, and the components of F have continuous partial derivatives, F is conservative. If f is a potential function of F , then f x = 2 xy ⇒ f ( x,y,z ) = x 2 y + g ( y,z ) , f y = x 2 + 2 yz ⇒ f ( x,y,z ) = x 2 y + y 2 z + h ( x,z ) , f z = y 2 ⇒ f ( x,y,z ) = y 2 z + k ( x,y ) for some functions g , h , and k of respective variables. By letting g ( y,z ) = y 2 z + C, h ( x,z ) = C, k ( x,y ) = x 2 y + C for an arbitrary constant C , one has f ( x,y,z ) = x 2 y + y 2 z + C. Page 2...
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 Fall '10
 Kwa
 Math, Calculus, Geometry, Derivative, 15 minutes, 2 K, Ohio State University, potential function, relevant supporting work

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