010.104-2006-2-fin - Honor Calculus2 Final Exam 2006 12 9{...

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Unformatted text preview: Honor Calculus2 Final Exam 2006 12 9{ 13r 15r 4 Z 9 < V: s2: M K+ c a Z @\ %V IU lǣ k k( 120). \ 1. (a) (7 pts) Let c(t) = (sin2 t2 , cos2 t2 , cos(2t2 )) (0 t 2 c (x + 3yz)ds. 2) be a path in space. Evaluate (b) (8 pts) Let C1 be a straight line from (0, 0) to (1, 1) and C2 be a portion of a circle x2 + y 2 = 2, x y from (1, 1) to (-1, -1). For F(x, y) = x2 i + (1 + y 2 )j, evaluate C1 +C2 F ds. 2. (10 pts) Evaluate the line integral (2xex + arctan2 y)dx + (sin3 y cos y + etan x )dy, C 2 where C is the boundary of the rectangle [0, ] [0, 1], oriented in the counterclockwise 4 direction. 3. (15 pts) The solid angle of the region R at the point P means the area of the projection of R into unit sphere centered at P . For example, the solid angle of the first octant 1 x > 0, y > 0, z > 0 at the origin is 8 4 = . 2 Let C be the cone given by z x2 + y 2 . Find the solid angle of C at the vertex of C. 4. (15 pts) Let S be a cylinder given by x2 + y 2 = 4 and 0 z 1. And let F(x, y, z) = (xy)i+(y+x sinh z)j+(y sinh z)k. (Note that sinh z = 1 (ez -e-z ).) Evaluate S curl FdS. 2 5. (15 pts) Let S be the union of S1 and S2 , where S1 = {(x, y, z)|z = x2 + y 2 , 1 z 2} and S2 = {(x, y, z)|x2 + y 2 + (z - 2)2 = 2, z 2}. For F = (zx + z 2 y + x)i + (z 3 yx + y 2 )j + z 4 x2 k, compute S ( F) dS. 6. (15 pts) Let W be a region x2 + y 2 1, 0 z 1 in space and W be the boundary of W with outward normal orientation. Evaluate W F dS for F(x, y, z) = (xy 2 + yz)i + (xz + yz)j + (x2 z + z 2 )k. 7. (a) (10 pts) Let F = (2xyez +6x)i+(x2 ez +z)j+(x2 yez +y)k. Prove that F is irrotational and find a function f such that f = F. (b) (10 pts) Let F = xi + yj - 2zk. Find G such that curl G = F. x 8. (15 pts) Let F = x2-y 2 i+ x2 +y2 j. For the closed curve C in Figure 1 with counter-clockwise +y orientation, compute C F ds. Figure 1: Figure for Problem 8 ...
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