16.9withScannedExamples

16.9withScannedExamples - 16.9: THE DIVERGENCE THEOREM KIAM...

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KIAM HEONG KWA 1. The Divergence Theorem A solid region E R 3 is said to be simple provided it is simultane- ously of types 1, 2, and 3, the definitions of which are given in section 15.6. In other words, if D 1 , D 2 , and D 3 are respectively the projections of E onto the xy -plane, the yz -plane, and the xz -plane, then a point ( x,y,z ) R 3 is in E if and only if ( x,y ) D 1 and u 1 ( x,y ) z u 2 ( x,y ), ( y,z ) D 2 and v 1 ( y,z ) x v 2 ( y,z ), and ( x,z ) D 3 and w 1 ( x,z ) y w 2 ( x,z ), where, for i = 1 , 2, u i ( x,y ), v i ( y,z ), and w i ( x,z ) are some continuous functions on D 1 , D 2 , and D 3 respectively. Suppose that E is a bounded simple solid region whose boundary surface S is given the positive (outward) orientation. Suppose also that F = P i + Q j + R k is a vector field on an open set containing E such that the partial derivatives of P , Q , and R are continuous. Then (1.1) ZZZ E ∇· E dV = ZZZ E ∂P ∂x dV + ZZZ E ∂Q ∂y dV + ZZZ E ∂R ∂z dV and (1.2) ZZ S F · d S = ZZ S P i · d S + ZZ S Q j · d S + ZZ S R k · d S . We would like to verify that ZZZ E ∂P ∂x dV = ZZ S P i · d S , (1.3a) ZZZ E ∂Q ∂y dV = ZZ S Q j · d S , (1.3b) ZZZ E ∂R ∂z dV = ZZ S R k · d S . (1.3c) Date : November 26, 2010. 1
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This note was uploaded on 11/11/2011 for the course MATH 254.01 taught by Professor Kwa during the Fall '10 term at Ohio State.

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16.9withScannedExamples - 16.9: THE DIVERGENCE THEOREM KIAM...

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