Math 227
Carter
Test 3
1. Consider the function f (x, y) = ex sin y + xy.
~ (15 points).
(a) Compute the gradient ∇f
(b) Compute the work done in moving a particle from (0, 0) to (ln 3, π/6) in the
gradient field. (15 points).
2. Use triple integrals (and cylindrical or spherical coordinates) to compute the volume
of a sphere of radius 6 centered at the origin. (15 points).
R
3. Use Green’s Theorem to evaluate Ω xy 2 dx + x3 dy where Ω is the rectangle with
vertices (0, 0), (2, 0), (2, 3), and (0, 3). (20 points).
4. Compute
Z
1
x dy − y dx
2 C
where C is the line from the point (a, b) to the point (c, d). (20 points).
5. Suppose that f is a real valued functions that is defined on an open, connected, region
~ is a continuous vector field in the same region. show that
of space and that G
~ · (f G)
~ = f (∇
~ · G)
~ +G
~ · ∇f
~
∇
where ∇· is the differential operator
∂
i
∂x
+
1
∂
j
∂y
+
∂
k
∂z
(15 points).