17.5 Ex. 24

17.5 Ex. 24 - 1160 C H A P T E R 17 L I N E A N D S U R FA...

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1160 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) 24. Calculate the fow rate through the upper hemisphere oF the sphere x 2 + y 2 + z 2 = R 2 ( z 0 ) For v as in Exercise 23. SOLUTION We use the parametrization, 8( θ , φ ) = ( R cos sin , R sin sin , R cos ) where, 0 < 2 π , 0 2 The normal vector is (see Eq. (4) in Section 17.4): n = T × T = R 2 sin - cos sin , sin sin , cos ® We write the velocity ±eld in terms oF the parameters: v = D x 2 + y 2 , 0 , z 2 E = D R 2 cos 2 sin 2 + R 2 sin 2 sin 2 , 0 , R 2 cos E = R 2 D sin 2 , 0 , cos E Hence, v · n = R 4 sin ³ sin 3 cos + cos 2 ´ = R 4 sin 4 cos + R 4 cos 2 sin The fow rate through the upper hemisphere S is equal to the fux oF the velocity vector across S .Thatis , ZZ S v · d S = Z / 2 0 Z 2 0 ³ R 4 sin 4 cos + R 4 cos 2 sin ´ d d = à R 4 Z 2 0 cos d ±Ã Z / 2 0 sin 4 d ± + R 4 · 2 Z / 2 0 cos 2 sin d = 0 + 2 R 2 · cos 4 4 ¯ ¯ ¯ ¯ ¯ / 2 = 0 = 2 R 4 µ 0 + 1 4 = R 4 2 Ft 3 / s 25. Calculate the fow rate oF a fuid with velocity ±eld v = - x , y , x 2 y ® (in Ft / s) through the portion oF the ellipse ³ x 2 ´ 2 + ³ y 3 ´ 2 = 1inthe xy -plane, where x , y 0, oriented with the normal in the positive z -direction. We use the Following parametrization For the surFace (see remark at the end oF the solution): 8 : x = 2 r cos , y = 3 r sin , z =
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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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