17.2 Ex 40-49

17.2 Ex 40-49 - 1084 C H A P T E R 17 L I N E A N D S U R...

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1084 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) 40. Let C be the path from P to Q in Figure 17 that traces C 1 , C 2 ,and C 3 in the orientation indicated. Suppose that Z C F · d s = 5 , Z C 1 F · d s = 8 , Z C 3 F · d s = 8 Determine: (a) Z C 3 F · d s( b ) Z C 2 F · d c ) Z C 1 C 3 F · d s x y P Q C 1 C 3 C 2 FIGURE 17 (d) What is the value of the line integral of F over the path that traverses the loop C 2 four times in the clockwise direction? SOLUTION x y P Q C 1 C 3 C 2 (a) If the orientation of the path is reversed, the line integral changes sign, thus: Z C 3 F · d s =− Z C 3 F · d s 8 (b) By additivity of line integrals, we have Z C F · d s = Z C 1 F · d s + Z C 2 F · d s + Z C 3 F · d s Substituting the given values we obtain 5 = 8 + Z C 2 F · d s + 8 or Z C 2 F · d s = 5 16 11 (c) Using properties of line integrals gives Z C 1 C 3 F · d s = Z C 1 F · d s + Z C 3 F · d s Z C 1 F · d s Z C 3 F · d s 8 8 16 (d) Using additivity and the integral over the curve with the reversed orientation, the line integral of F over the path that traverses the loop C 2 four times in the clockwise direction is: 4 Z C 2 F · d s = 4 · µ Z C 2 F · d s 4 Z C 2 F · d s 4 · ( 11 ) = 44 In Exercises 41–44, let F be the vortex vector feld (so-called because it swirls around the origin as shown in Figure 18) F = ¿ y x 2 + y 2 , x x 2 + y 2 À
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SECTION 17.2 Line Integrals (ET Section 16.2) 1085 FIGURE 18 Vector feld F = ¿ y x 2 + y 2 , x x 2 + y 2 À . 41. Let I = Z C F · d s ,where C is the circle oF radius 2 centered at the origin oriented counterclockwise (±igure 18). (a) Do you expect I to be positive, negative, or zero? (b) Evaluate I . (c) VeriFy that I changes sign when C is oriented in the clockwise direction. SOLUTION (a) When the circle is oriented counterclockwise, the dot product oF F with the unit tangent vector at each point along the circle is positive. ThereFore, we expect the vector line integral I to be positive. (b) 2 x y P The circle oF radius 2 oriented counterclockwise has the parametrization: c ( t ) = ( 2cos t , 2sin t ), 0 t 2 π Hence, F ( c ( t )) = ¿ t 4cos 2 t + 4sin 2 t , t 2 t + 2 t À = 1 2 h− sin t , cos t i c 0 ( t ) = h− t , t i ThereFore, the integrand is the dot product, F ( c ( t )) · c 0 ( t ) = 1 2 h− sin t , cos t i · h− t , t i = sin 2 t + cos 2 t = 1 We obtain the Following integral: Z C F · d s = Z 2 0 F ( c ( t )) · c 0 ( t ) dt = Z 2 0 1 = 2 (c) When C is oriented in the clockwise direction, the parameter t is changing From 2
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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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17.2 Ex 40-49 - 1084 C H A P T E R 17 L I N E A N D S U R...

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