17.5 Ex 18-21

17.5 Ex 18-21 - S E C T I O N 17.5 Surface Integrals of Vector Fields(ET Section 16.5 1155 Step 1 Compute the tangent and normal vectors We have Tu

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SECTION 17.5 Surface Integrals of Vector Fields (ET Section 16.5) 1155 Step 1. Compute the tangent and normal vectors. We have, T u = ∂8 u = u ³ u 3 v, u + v, v 2 ´ = D 3 u 2 , 1 , 0 E T v = ∂v = ³ u 3 u + 2 ´ = h− 1 , 1 , 2 v i T u × T v = ¯ ¯ ¯ ¯ ¯ ¯ ij k 3 u 2 10 112 v ¯ ¯ ¯ ¯ ¯ ¯ = ( 2 v) i ³ 6 u 2 v ´ j + ³ 3 u 2 + 1 ´ k = D 2 6 u 2 3 u 2 + 1 E Since the normal is pointing downward, the z -coordinate is negative, hence, n = D 2 6 u 2 3 u 2 1 E Step 2. Evaluate the dot product F · n . We frst express F in terms oF the parameters: F (8( u ,v)) = h y , z , 0 i = D u + 2 , 0 E We compute the dot product: F (8( u · n ( u ,v) = D u + 2 , 0 E · D 2 6 u 2 3 u 2 1 E =− 2 v( u + + 6 u 2 v · v 2 + 0 2 v u 2 v 2 + 6 u 2 v 3 Step 3. Evaluate the surFace integral. The surFace integral is equal to the Following double integral: ZZ S F · d S = D F (8( u · n ( u dud v = Z 3 0 Z 2 0 ³ 2 u v 2 v 2 + 6 u 2 v 3 ´ v = Z 3 0 u 2 v 2 v 2 u + 2 u 3 v 3 ¯ ¯ ¯ ¯ 2 u = 0 d v = Z 3 0 ³ 16 v 3 4 v 2 4 v ´ d v = 4 v 4 4 3 v 3 2 v 2 ¯ ¯ ¯ ¯ 3 0 = 270 18. Let S be the oriented halF-cylinder in ±igure 14. In (a)–(F), determine whether S F · d S is positive, neg- ative, or zero. Explain your reasoning. (a) F = i( b ) F = j (c) F = k( d ) F = y i (e) F y j (f) F = x j y x n z FIGURE 14 SOLUTION
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1156 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) y x n 3 1 z 1 S is parametrized by: 8( θ , z ) = ( cos , sin , z ), 0 z 3 , π 2 2 Hence, T = ∂8 = h− sin , cos , 0 i T z = z = h 0 , 0 , 1 i T × T z = ¯ ¯ ¯ ¯ ¯ ¯ ij k sin cos 0 00 1 ¯ ¯ ¯ ¯ ¯ ¯ = ( cos ) i + ( sin ) j = h cos , sin , 0 i The normal to S is pointing in the outward direction, hence the x -coordinate of n is positive. Since 2 2 ,we have cos 0, hence, n = h cos , sin , 0 i (a) Since F · n = h 1 , 0 , 0 i · h cos , sin , 0 i = cos ,and
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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 18-21 - S E C T I O N 17.5 Surface Integrals of Vector Fields(ET Section 16.5 1155 Step 1 Compute the tangent and normal vectors We have Tu

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