Summary: 18.024ESG Exam 4
Pramod N. Achar
1. Let f : R3
be the vector field f(x, y, z) = yi + zj + xk. Let S be the portion of the surface given
by z = 1 - x2
lying above the xy-plane.
(a) Compute the vector field g = curl f.
(b) Using a parametrization of S, convert the surface integral S
g into a double integral over some
region in R2
. Set up the limits on this double integral, and simplify the integrand as much as
possible, but do not evaluate it.
(c) Compute S g by using Stokes' Theorem to convert it into a line integral.
2. Let S be the region in R2
bounded by the curves xy = 1, xy = 2, y = x, and y = 4x. Let f : R R
be a scalar function of one variable. Using the change of variables x = u/v, y =