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17.5 Ex 8-9 - 1148 C H A P T E R 17 L I N E A N D S U R FA...

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1148 C H A P T E R 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) 8. F = x , y , z , part of sphere x 2 + y 2 + z 2 = 1, where 1 2 z 3 2 , inward-pointing normal SOLUTION z x y We parametrize S by the following parametrization: ( θ , φ ) = ( cos θ sin φ , sin θ sin φ , cos φ ) D : 0 θ 2 π , φ 0 φ φ 1 f 0 1 3 2 The angles φ 0 and φ 1 are determined by: cos φ 0 = 3 2 φ 0 = π 6 cos φ 1 = 1 2 φ 1 = π 3 f 1 1 1 2 Step 1. Determine the normal vector. The normal vector pointing to the outside of the sphere is: n = T φ × T θ = sin φ cos θ sin φ , sin θ sin φ , cos φ (Notice that for π 6 φ π 3 , sin φ cos φ > 0, therefore the z -component is positive and the normal points to the outside of the sphere). Step 2. Evaluate the dot product F · n . We express F in terms of the parameters: F ( ( θ , φ ) ) = x , y , z = ( cos θ sin φ , sin θ sin φ , cos φ ) Hence: F ( ( θ , φ ) ) · n ( θ , φ ) = cos θ sin φ , sin θ sin φ , cos φ · sin φ cos θ sin φ , sin θ sin φ , cos φ = sin φ cos 2 φ sin 2 φ + sin 2 θ sin 2 φ + cos 2 φ = sin φ · 1 = sin
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