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234-shi-06fafin

# 234-shi-06fafin - FINAL EXAM MATH 234 CALCULUS ~ FUNCTIONS...

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Unformatted text preview: FINAL EXAM, MATH 234: CALCULUS ~_ FUNCTIONS OF SEVERAL VARIABLES DECEMBER 20, 2006 No calculators, books or papers may be used. This examination consists of ﬁve questions, Each problem is worth twenty points. Partial credits will be given only When a substantial part of a problem has been worked out. Merely displaying some formulas is not sufﬁcient ground for receiving partial Credits. PLEASE BOX YOUR ANSWERS. Hint: Use of Stokes’ and divergence theorems may be needed for solving ’ some of the questions. 0 YOUR NAME, PRINTED: 0 YOUR LECTURE SECTION, (CIRCLE ONE): HUANG OZMAN ‘ J ORSTAD HA‘ACK SHI Name: ' ’ 1 // curl (Fm dS, G F: xf+ eyf+zsin E, and G’ is the part of the sphere x2 + 3/2 + 2.2 = 2 above the plane z = 1 and FL is the upward unit normal. ’ 1. Evaluate where Name: 2. Calculate // F-ﬁdS as on the solid given, by x2 + y2 + 22 S 17 where' F = (‘23: + yz)i‘+ 3yf+ zE, and BS denotes the boundary of S. V Name: 3. Find the tangent plane to the surfaCe m2 + y2 + 22 = 16 atx=2,y;3,2=\/§. Name: 4 4. Find a unit vector in the direction which makes f (x, y) = x3 — 3/5 increase most rapidly at F = (2, —1). Name: > 5 5. (a) Show that the line integral ‘/(y2 + 2mg) da: + (x2 + dy C is path independent by ﬁnding a potential function f. (b) Use the potential function in (a) to evaluate the integral where C is any curve joining (0,0) to (1,1). ...
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