16.9withScannedExamples

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 8

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online