Unformatted text preview: MATH 241 881 Calculus III FINAL Huaizhong Ren July 25, 2003 INSTRUCTIONS: Work all the 4 problems on the 4 answer sheets provided. Each problem must be on a separate answer sheet. Be sure to put your name and the problem number on each sheet.
Show all your work to get full credit. No graphing calculators allowed. 1. Evaluate the following line integrals (a) (15 pts) fC($ + y)ds, Where C is the triangle with the vertices 0(0, 0), A(1, 0), and B(0,1). (b) (15 pts) fC[(a: + y)dx + (:12 -— y)dy]/(3r2 + 312), where C is the circle 2:2 + y2 = R2 oriented
counterclockwise. 2. Evaluate the following surface integrals. (a) (15 pts) ffz(x2 + y2)dS, where Z is given by z = 2 — (a:2 + 3/2), 2 _>_ 0.
(b) (15 pts) ffEF ~ ndS, where F = (cc + 1)i + yj + k, and E is the upper half of the sphere 2:2 + y2 + 22 = 4, with the upward normal vector n. 3. Answer the following questions. Show evidence with your answer. (a) (10 pts) Let F = yzi + xzj + xyk. Is the integral f0 F - dr independent of the path? Evaluate
the integral. (b) (10 pts) Let F = -—yi/(:I;2 + y?) + :rj/(x2 + yz). Show that curlF = 0 but fCF - dr 76 0 for
C' : x2 + y2 = R2, oriented counterclockwiseDoes that contradict the Green’s Thoerem? 4. Use Stokes Theorem or the Divergence Theorem to evaluate the following integrals. (a) (10 pts) fCF - dr, where F = (z — y)i + (x — z)j + (y —— 2:)k, and C is the intersection of the
sphere 3:2 + y2 + .22 = 5 with the plane 2 = ~2, oriented counterclockwise viewed from above. (b) (10 pts) f fEF - ndS, where F = $3i + y3j + 23k and 2 is the lower half of the unit sphere
m2 + y2 + 22 = 1 with the downward normal vector n. ...
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