div_2 - ) , , ( z y x F ! , F g ! ! = . Then g(x,y,z) is...

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PHY2061 R. D. Field Department of Physics div_2.doc Univesity of Florida Divergence Theorem The divergence theorem states that the integral of the divergence of a vector function over a volume, V, is equal to the flux, Φ F , of the vector function through the closed surface, S, that encloses the volume V: = S V A d F dV F ! ! ! ! ) ( Proof ( sketch ): = = = Φ = = V V N N i S i i i N i S i S F dV F A d F V V A d F A d F i i i ) ( 1 0 1 1 ! ! ! ! ! ! ! ! The Laplacian Operator Suppose that the vector function, ) , , ( z y x F ! , is the gradient of the scalar function f(x,y,z), f F = ! ! . Now suppose we construct a new scalar function g(x,y,z) that is the divergence of
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Unformatted text preview: ) , , ( z y x F ! , F g ! ! = . Then g(x,y,z) is the divergence of the gradient of f(x,y,z) as follows: f f g 2 = = ! ! In cartesian (or retangular) coordinates: z z y y x x + + = ! 2 = ! ! 2 2 2 2 2 2 2 z y x + + = 2 2 2 2 2 2 2 z f y f x f f g + + = = Closed Surface S Volume V enclosed by surface S Laplacian Operator Scalar f(x,y,z) Laplacian Operator Scalar g(x,y,z) Divergence Theorem...
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This note was uploaded on 05/31/2011 for the course PHY 2061 taught by Professor Fry during the Spring '08 term at University of Florida.

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