Spring 2007 - Takeda's Class - Exam 2

Spring 2007 - Takeda's Class - Exam 2 - Math 20E, Midterm 2...

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Unformatted text preview: Math 20E, Midterm 2 Solutions June 8, 2007 Name : PID : TA : Sec. No : Sec. Time : This exam consists of 6 pages including this front page. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justification will receive no credit. 3. You may use one 4-by-6 index card, both sides. 4. Show your ID on your desk. Score 1 10 2 10 3 10 4 10 5 10 Total 50 1 1. (a) Compute Z c y ds, where c is the path given by c ( t ) = 7 i + t j + t 2 k , 0 t 1. Answer : Note that c ( t ) = j + 2 t k and so || c ( t ) || = 1 + 4 t 2 . So Z c y ds = Z 1 t 1 + 4 t 2 dt. Let 1 + 4 t 2 = u . Then Z 1 t 1 + 4 t 2 dt = Z 5 1 1 8 u du = 1 8 2 3 u 3 2 i 5 1 = 5 3 / 2- 1 12 . (b) Verify that the path c ( t ) = (sin t, cos t, e 2 t ) is a flow line of the vector field F = ( y,- x, 2 z ). Answer : c ( t ) = (cos t,- sin t, 2 e 2 t ) . Also F ( c ( t )) = (cos t,- sin t, 2 e 2 t ) ....
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Spring 2007 - Takeda's Class - Exam 2 - Math 20E, Midterm 2...

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