Vector Calc Practice Midterm-Answers 20E 1

Vector Calc Practice Midterm-Answers 20E 1 - Math 20E....

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Unformatted text preview: Math 20E. Midterm 1. April 25, 2004 ANSWER ALL 5 QUESTIONS. Each question is worth 20 points. 1. Let C be the triangle with vertices (1 , , 0), (0 , 1 , 0) and (0 , , 2), oriented so that when you trace out the triangle, you pass through the points in the order given. Compute the circulation R C F d R where F = x i + z j- y k . Solution 1 . The curve consists of 3 line segments: C 1 joins (1 , , 0) to (0 , 1 , 0). C 2 joins (0 , 1 , 0) to (0 , , 2). C 3 joins (0 , , 2) to (1 , , 0). We can parameterize by C 1 : ( x,y,z ) = (1- t,t, 0) C 2 : ( x,y,z ) = (0 , 1- t, 2 t ) C 3 : ( x,y,z ) = ( t, , 2- 2 t ). Then Z C F d R = Z C xdx + zdy- ydz. We can save ourselves some work by noticing that xdx = d ( x 2 ) / 2 is an exact differential and since C is closed, this will not contribute to the answer. Hence Z C F d R = Z C zdy- ydz = Z C 1 zdy- ydz + Z C 2 zdy- ydz + Z C 3 zdy- ydz = Z 1 dt + Z 1 2 t (- 1) dt- (1- t )2 dt + Z 1 dt = Z 1 (- 2 t- 2 + 2 t ) dt = Z 1- 2 dt =- 2 ....
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This note was uploaded on 04/07/2008 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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Vector Calc Practice Midterm-Answers 20E 1 - Math 20E....

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