Math 251
Chapter 14 Sample Exam Problems
May 3, 2011
1. Let f (x, y, z) = z sin(xy). Find div(∇f ).
(A) z(x + y) cos(xy) + sin(xy)
(B) −z(x2 + y 2 ) sin(xy)
(C) hyz cos(xy), xz cos(xy), sin(xy)i
(D) h−y 2 z sin(xy), −x2 z sin(xy), 0i
(E) 0
2. Z
Let C be the curve in the xy-plane given by x = 3t, y = 2t2 , 0 ≤ t ≤ 1. Evaluate
3y dx − x2 dy.
C
(A) −3
(B) −2
(C) 0
(D) 1
(E) 2
3. Let C be the curve defined by parametric equations x = −t cos t, y = t sin t, z = tet/π
with 0 ≤ t ≤ π. Let F(x, y, z) = h1 + yz, 1 + xz, 1 + xyi. Compute
Z
F · dr.
C
Hint: F is a conservative vector field—find a potential function for F and use independence of path.
(A) π + e + πe
(C) π + πe
(C) π + πe + π 2 e
(D) π 2 + πe + πe2
(E) 0
1
4. Let S be the portion of the paraboloid z = 1 − x2 − y 2 above the xy-plane, and let n
be an upward unit normal
Z Z vector for S. Let F(x, y, z) = hx, y, 2zi. What is the value
F · n dS?
of the surface integral
S
(A) 2π
(B) −2π
(C) π
(D) −π
(E) 0
5. Evaluate the following line integrals.
Z
(a) Evaluate (x2 + y 2 + z 2 ) ds, where C is the curve in 3-space given by x = 4 cos t,
C
y = 4 sin t, z = 3t, 0 ≤ t ≤ 2π.
Z
2
(b) Evaluate
3yex + 2ex dx +
C
3 2
x
2
+ sin y dy, where C is the triangle in the xy-
plane with vertices (0, 0), (2, 0), (2, 4), oriented counter-clockwise. (Hint: Use Green’s
Theorem.)
6. Let F = 2yi − 2xj be a vector field in the xy-plane. Let C be the unit circle oriented
counterclockwise, and let R be the closed unit disk.
I
(a) According to Green’s Theorem,
F · dr is equal to what double integral?
C
I
F · dr directly.
(b) Evaluate
C
(c) Evaluate the double integral you provided in part (a) directly.
ZZ
7. Use Stokes’ Theorem to calculate the surface integral
(curl F) · n dS, where S is
S
the part of the paraboloid z = 9 − x2 − y 2 above the xy-plane, n is the outward unit
normal, and F = hz + y, z − x, 2zi.
8. Let S be the graph of f (x, y) = x2 + y 2 above the unit disk R in the xy-plane. The
boundary of S is the circle C that is a translate of the usual unit circle in the xy-plane,
1 unit directly upward.
Let F = yi − xj + yzk. Use Stokes’ Theorem to evaluate
I
F · dr,
C
where C is oriented counter-clockwise.
2