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Vector Calc Practice Midterm-Answers 20E 1

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

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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), (0 , 1 , 0) and (0 , 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 , 0) to (0 , 1 , 0). C 2 joins (0 , 1 , 0) to (0 , 0 , 2). C 3 joins (0 , 0 , 2) to (1 , 0 , 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, 0 , 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 0 0 dt + Z 1 0 2 t ( - 1) dt - (1 - t )2 dt + Z 1 0 0 dt = Z 1 0 ( - 2 t - 2 + 2 t ) dt = Z 1 0 - 2 dt = - 2 . 2. For the following vector fields F , either find a scalar field φ such that F = φ , or show that F is not conservative on R 3 .

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

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