17.5 Prel 1-6, Ex 1-2

17.5 Prel 1-6, Ex 1-2 - S E C T I O N 17.5 Surface...

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SECTION 17.5 Surface Integrals of Vector Fields (ET Section 16.5) 1141 Using the Surface Integral over a Graph we have: Area ( S ) = ZZ S 1 dS = D q 1 + g 2 x + g 2 y dA (1) In parametrizing the surface by φ ( x , y ) = ( x , y , g ( x , y )) , ( x , y ) = D ,wehave: T x = ∂8 x = h 1 , 0 , g x i T y = y = - 0 , 1 , g y ® Hence, n = T x × T y = ¯ ¯ ¯ ¯ ¯ ¯ ij k 10 g x 01 g y ¯ ¯ ¯ ¯ ¯ ¯ =− g x i g y j + k = - g x , g y , 1 ® k n k= q g 2 x + g 2 y + 1 n k k There are two adjacent angles between the normal n and the vertical, and the cosines of these angles are opposite numbers. Therefore we take the absolute value of cos to obtain a positive value for Area ( S ) . Using the Formula for the cosine of the angle between two vectors we get: | cos |= | n · k | k n kk k k = | - g x , g y , 1 ® · h 0 , 0 , 1 i | q 1 + g 2 x + g 2 y · 1 = 1 q 1 + g 2 x + g 2 y Substituting in (1) we get: Area ( S ) = D | cos | 17.5 Surface Integrals of Vector Fields (ET Section 16.5) Preliminary Questions 1. Let F be a vector ±eld and 8( u ,v) a parametrization of a surface S ,andset n = T u × T v . Which of the following is the normal component of F ? (a) F · n( b ) F · e n SOLUTION The normal component of F is F · e n rather than F · n . 2. The vector surface integral S F · d S is equal to the scalar surface integral of the function (choose the correct answer): (a) k F k (b) F · n ,where n is a normal vector (c) F · e n e n is the unit normal vector The vector surface integral R S F · d S is de±ned as the scalar surface integral of the normal component of F on the oriented surface. That is, R S F · d S = R S ( F · e n ) as stated in (c). 3. S F · d S is zero if (choose the correct answer): (a) F is tangent to S at every point. (b) F is perpendicular to S at every point. Since R S F · d S is equal to the scalar surface integral of the normal component of F on S , this integral is zero when the normal component is zero at every point, that is, when F is tangent to S at every point as stated in (a).
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1142 CHAPTER 17 LINE AND SURFACE INTEGRALS
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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 Prel 1-6, Ex 1-2 - S E C T I O N 17.5 Surface...

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