MATH 53 PRACTICE FINAL EXAM
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 .
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
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