18.3 Prel Q. Ex 1 - 11

18.3 Prel Q. Ex 1 - 11 - S E C T I O N 18.3 Divergence...

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SECTION 18.3 Divergence Theorem (ET Section 17.3) 1245 (b) By Stokes’ Theorem, Z C r F · d s = ZZ S r curl ( F ) · d S By part(a) we have m ( r ) 1 π r 2 Z C r F · d s M ( r ) (2) We take the limit over the circles of radius r centered at Q ,as r 0. As r 0, the regions S r are approaching the center Q . The continuity of the curl implies that lim r 0 m ( r ) = lim r 0 M ( r ) = curl ( F )( Q ) · e n ( Q ) = curl ( F )( Q ) · e Therefore, lim r 0 m ( r ) lim r 0 1 r 2 Z C r F · d s lim r 0 M ( r ) curl ( F )( Q ) · e lim r 0 1 r 2 Z C r F · d s curl ( F )( Q ) · e Hence, lim r 0 1 r 2 Z C r F · d s = curl ( F )( Q ) · e 18.3 Divergence Theorem (ET Section 17.3) Preliminary Questions 1. What is the Fux of F = h 1 , 0 , 0 i through a closed surface? SOLUTION The divergence of F = h 1 , 0 , 0 i is div ( F ) = P x + Q y + R z = 0, therefore the Divergence Theorem implies that the Fux of F through a closed surface S is S F · d S = ZZZ W div ( F ) dV = W 0 = 0 2. Justify the following statement: The Fux of F = ± x 3 , y 3 , z 3 ® through every closed surface is positive. The divergence of F = D x 3 , y 3 , z 3 E is div ( F ) = 3 x 2 + 3 y 2 + 3 z 2 Therefore, by the Divergence Theorem, the Fux of F through a closed surface S is S F · d S = W ( 3 x 2 + 3 y 2 + 3 z 2 ) Since the integrand is positive for all ( x , y , z ) 6= ( 0 , 0 , 0 ) , the triple integral, hence also the Fux, is positive. 3. Which of the following expressions are meaningful (where F is a vector ±eld and ϕ is a function)? Of those that are meaningful, which are automatically zero? (a) div ( ) (b) curl ( ) (c) curl ( ) (d) div ( curl ( F )) (e) curl ( div ( F )) (f) ( div ( F )) (a) The divergence is de±ned on vector ±elds. The gradient is a vector ±eld, hence div ( ) is de±ned. It is not automat- ically zero since for = x 2 + y 2 + z 2 we have div ( ) = div h 2 x , 2 y , 2 z i = 2 + 2 + 2 = 6 0 (b) The curl acts on vector valued functions, and is such a function. Therefore, curl ( ) is de±ned. Since the gradient ±eld is conservative, the cross partials of are equal, or equivalently, curl ( ) is the zero vector. (c) The curl is de±ned on vector ±elds rather than on scalar functions. Therefore, curl ( ) is unde±ned. Obviously, curl ( ) is also unde±ned.
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1246 CHAPTER 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS (ET CHAPTER 17) (d) The curl is defned on the vector feld F and the divergence is defned on the vector feld curl ( F ) . ThereFore the expression div ( curl ( F )) is meaningFul. We show that this vector is automatically zero: div ( curl ( F )) = div ¿ F 3 y F 2 z , F 1 z F 3 x , F 2 x F 1 y À = x µ F 3 y F 2 z + y µ F 1 z F 3 x + z µ F 2 x F 1 y = 2 F 3 x y 2 F 2 x z + 2 F 1 y z 2 F 3 y x + 2 F 2 z x 2 F 1 z y = Ã 2 F 3 x y 2 F 3 y x !
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This homework help was uploaded on 04/13/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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18.3 Prel Q. Ex 1 - 11 - S E C T I O N 18.3 Divergence...

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