Math 317, Fall 2013, Section 101
Page 1 of 7
Midterm Exam II
October 30, 2013
No books. No notes. No calculators. No electronic devices of any kind.
Name
Student Number
Problem 1. (5 points)
~ = F~ , where F~ is the vector field
(a) Find a function f (x, y, z) such that ∇f
F~ (x, y, z) = hyz + 2y, xz + 2x, xy + 6zi.
~ × F~ .
(b) Compute ∇
Math 317, Midterm Exam II
Page 2 of 7
1
2
3
4
5
total/25
Problem 2. (5 points)
Let C be √
the part of the parabola y 2 = x + 2 in the second quadrant, starting at the
point (0, 2) and ending at the point (−2, 0). So C is not a closed curve. Compute
the line integral
Z
h5y, 2xi · d~r
C
Math 317, Midterm Exam II
Page 3 of 7
Problem 3. (5 points)
Let C be the circle of radius 2 centered at the point (1, 1) in the xy-plane, oriented
counterclockwise. Use Green’s theorem to compute
I
F~ · d~r
C
where F~ = h5x − y + 2xy, 2x + x2 + 3yi. If you do this problem correctly, there is
no need to compute and integral.
Math 317, Midterm Exam II
Problem 4. (5 points)
Compute the curl and the divergence of the vector field
F~ = hcos(xy), ez+y , tan(x + z)i
Page 4 of 7
Math 317, Midterm Exam II
Page 5 of 7
Problem 5. (5 points)
(a) Verify that the vector field in the xy-plane given by F~ = hx2 , ey cos yi is
conservative.
(b) Let C be the half ellipse parametrized by
~r(t) = h2 cos t, sin ti
0≤t≤π .
R
Compute the line integral C F~ · d~r using path independence.
Math 317, Midterm Exam II
Overflow space I.
Page 6 of 7
Math 317, Midterm Exam II
Overflow space II.
Page 7 of 7