17.4 Prel 1-6, Ex 1-3

17.4 Prel 1-6, Ex 1-3 - 1106 C H A P T E R 17 L I N E A N D...

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1106 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) Further Insights and Challenges 25. The vector feld F = ¿ x x 2 + y 2 , y x 2 + y 2 À is defned on the domain D ={ ( x , y ) 6= ( 0 , 0 ) } . (a) Show that F satisfes the cross-partials condition on D . (b) Show that ϕ ( x , y ) = 1 2 ln ( x 2 + y 2 ) is a potential Function For F . (c) Is D simply connected? (d) Do these results contradict Theorem 4? SOLUTION (a) We compute the partials oF F : F 2 x = x µ y x 2 + y 2 = 2 xy ( x 2 + y 2 ) 2 F 1 y = y µ x x 2 + y 2 = 2 yx ( x 2 + y 2 ) 2 The cross partials are equal in D . (b) We compute the gradient oF ( x , y ) = 1 2 ln ³ x 2 + y 2 ´ : = ¿ x , x À = 1 2 ¿ 2 x x 2 + y 2 , 2 y x 2 + y 2 À = ¿ x x 2 + y 2 , y x 2 + y 2 À = F (c) D is not simply-connected since it has a “hole” at the origin. (d) The requirement in Theorem 4 (that the domain be simply connected) is a suFfcient condition For a vector feld with equal cross-partials to have a potential Function. It is not necessary, since as in our example, even iF the domain is not simply-connected the feld may have a gradient Function. Moreover, For any closed curve in D , havethesameva lue aFter completing one round along c . This is perhaps best seen by noting that = log ( r ) in polar coordinates, which will be independent oF θ . ThereFore, Z c F · d s = 0 Hence, F is conservative. PQ 17.4 Parametrized Surfaces and Surface Integrals (ET Section 16.4) Preliminary Questions 1. What is the surFace integral oF the Function f ( x , y , z ) = 10 over a surFace oF total area 5? Using SurFace Integral and SurFace Area we have: ZZ S f ( x , y , z ) dS = D f (8( u ,v)) k n ( u ,v) k dud v = D 10 k n ( u k v = 10 D k n ( u k v = 10 Area ( S ) = 10 · 5 = 50 2. What interpretation can we give to the length k n k oF the normal vector For a parametrization 8( u ?
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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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17.4 Prel 1-6, Ex 1-3 - 1106 C H A P T E R 17 L I N E A N D...

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