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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 C H A P T E R 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) 24. Calculate the flow 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, ( θ , φ ) = ( R cos θ sin φ , R sin θ sin φ , R cos φ ) where, 0 θ < 2 π , 0 φ π 2 The normal vector is (see Eq. (4) in Section 17.4): n = T φ × T θ = R 2 sin φ cos θ sin φ , sin θ sin φ , cos φ We write the velocity field in terms of the parameters: v = x 2 + y 2 , 0 , z 2 = R 2 cos 2 θ sin 2 φ + R 2 sin 2 θ sin 2 φ , 0 , R 2 cos φ = R 2 sin 2 φ , 0 , cos φ Hence, v · n = R 4 sin φ sin 3 φ cos θ + cos 2 φ = R 4 sin 4 φ cos θ + R 4 cos 2 φ sin φ The flow rate through the upper hemisphere S is equal to the flux of the velocity vector across S . That is, S v · d S = π / 2 0 2 π 0 R 4 sin 4 φ cos θ + R 4 cos 2 φ sin φ d θ d φ = R 4 2 π 0 cos θ d θ π / 2 0 sin 4 φ d φ + R 4 · 2 π π / 2 0 cos 2 φ sin φ d φ = 0 + 2 π R 2 · cos 4 φ 4 π / 2 φ = 0 = 2 π R 4 0 + 1 4 = π R 4 2 ft 3 / s 25. Calculate the flow rate of a fluid with velocity field v = x , y , x 2 y (in ft / s) through the portion of the ellipse x 2 2 + y 3 2 = 1 in the xy -plane, where x , y 0, oriented with the normal in the positive
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