Spring 2007 - Takeda's Class - Final Exam

Spring 2007 - Takeda's Class - Final Exam - Math 20E Final...

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Unformatted text preview: Math 20E, Final Exam Solutions June 15, 2007 Name : PID : TA : Sec. No : Sec. Time : This exam consists of 11 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 two 4-by-6 index cards, both sides. 4. You have two hours for this exam. Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1 1. Let F = x 2 i- xe y j + 2 xyz k . (a) Find div F . Answer : div F = 2 x- xe y + 2 xy. (b) Find curl F . Answer : curl F = i j k ∂ ∂x ∂ ∂y ∂ ∂z x 2- xe y 2 xyz = ∂ ∂y ∂ ∂z- xe y 2 xyz i- ∂ ∂x ∂ ∂z x 2 2 xyz j + ∂ ∂x ∂ ∂y x 2- xe y k = 2 xz i- 2 yz j- e y k . 2 2. (a) Express the paraboloid z = x 2 + y 2 using the cylindrical coordinates. Answer : Note that in the cylindrical coordinates, x = r cos θ, y = r sin θ, z = z. Hence z = x 2 + y 2 is z = r 2 . (b) Show that the two surfaces z = xy and z = 3 4 x 2- y 2 intersect perpen- dicularly at the point (2 , 1 , 2). (Hint: If two surfaces intersect perpen- dicularly at a point, then the normal vectors on the two surfaces at the point are perpendicular to each other.) Answer : Let f ( x, y, z ) = z- xy and g ( x, y, z ) = z- 3 4 x 2 + y 2 ....
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Spring 2007 - Takeda's Class - Final Exam - Math 20E Final...

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