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