17.5 Ex 8-9

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 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) 8. F = h x , y , z i , 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: 8( θ , φ ) = ( cos sin , sin sin , cos ) D : 0 2 π , 0 1 f 0 1 3 2 The angles 0 and 1 are determined by: cos 0 = 3 2 0 = 6 cos 1 = 1 2 1 = 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 × T = sin - cos sin , sin sin , cos ® (Notice that for 6 3 ,sin cos > 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 ( , ) ) = h x , y , z i = ( cos sin , sin sin , cos ) Hence: F ( , ) ) · n ( , ) = - cos sin , sin sin , cos ® · sin - cos sin , sin sin , cos ® = sin ³ cos 2 sin 2 + sin 2 sin 2 + cos 2 ´ = sin · 1 = sin Step 3. Evaluate the surface integral. We have: ZZ S F · d S = D F ( , ) ) · n ( , ) d d = Z 2 0 Z / 3 / 6 sin d d = 2 Z / 3 / 6 sin d = 2 ( cos ) ¯ ¯ ¯ ¯ / 3 = / 6 = 2 ³ cos 3 + cos 6 ´ = 2 Ã
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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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17.5 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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