Mock Final

Topics: Analytic geometry, Trigraph, Line Pages: 2 (553 words) Published: December 29, 2011
MATH 53 PRACTICE FINAL EXAM
JAMES MCIVOR

The material included here does not necessarily reflect what will be on the actual exam. Make sure to go over the review problems Professor Frenkel posted on his webpage. I recommend first looking over all the problems and deciding which strategy to use, before actually calculating anything. (1) Let S be the part of the plane 2x + 3y + 4z = 12 that lies in the first octant, with upward orientation. Compute the surface integral S F · dS, where F = xi + yj + zk. (2) Let F be the vector field x2 + y 2 , xy + z, z 2 and C be the curve of intersection of the sphere x2 + y 2 + z 2 = 4 and the plane x = 1, oriented counter-clockwise when viewed from the positive x-axis. Compute C F · dr. (3) Let S be the surface consisting of the two spheres x2 + y 2 + z 2 = 1 (oriented inward) and 2 x2 + y 2 + z 2 = 4 (oriented outward), and let F = e2yz i + yzj − sin(x2 y)k. find S F · dS. (4) Find the mass of the cylindrical region x2 + z 2 = 1, 0 ≤ y ≤ 3, if its density is given by √ ρ(x, y, z) = y x2 + z 2 . (5) Mark each of the following as True or False y x (a) If F = 2 i− 2 j, then F · dr = F · dr, where C1 is the straight-line x + y2 x + y2 C1 C2 path from (−1, −1) to (1, −1) to (1, 1), and C2 is the straight-line path from (−1, −1) to (−1, 1) to (1, 1). (b) If S is the hemisphere x2 + y 2 + z 2 = 1 with y ≥ 0, oriented towards the positive y-axis, and C is the curve parametrized by cos t, 0 sin t , 0 ≤ t ≤ 2π, then Curl F · dS = S C

F · dr

for any F whose components have continuous partial derivatives on R3 . (c) A vector field P i + Qj + Rk on R3 is conservative if there is a function f (x, y, z) such that ∂f = P , ∂f = Q, and ∂f = R. ∂x ∂y ∂z 3 (d) There is a vector field F on R3 such that Curl F = 2x2 y + 3z, xy 2 − 3yz, 2 z 2 − 5xyz .

(e)

x2 y dx + xy 2 dy = C2 x2 y dx + xy 2 dy, where C1 , C2 are circles in R2 of radius one and two, respectively, both centered at (0, 0) and oriented counterclockwise. C1

(f)
C...
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