Math 52 - Winter 2007 - Final Exam
Name:
Student ID:
Section number and TA name:
Signature:
Instructions: Print your name and student ID number, print your section number and TA’s
name, write your signature to indicate that you accept the honor code. During the test, you
may not use notes, books, calculators or telephones. Read each question carefully, and show
all your work. Put a box around your final answer to each question. There are 9 questions.
Point values are given in parentheses. You have 3 hours to answer all the questions.
Question
Score
Maximum
1
20
2
8
3
10
4
12
5
10
6
10
7
10
8
10
9
10
Total
100
Problem 1. Mark the statements below as TRUE or FALSE. Each correct answer: +2 pts.
Each wrong or missing answer: 0 pt.
In all the statements below, 1) the letter f denotes a scalar field, three times differentiable
where it is defined; 2) the letter F denotes a vector field, twice differentiable where it is
defined; 3) the letter D denotes a bounded region in R3 , whose boundary ∂D is the union of
finitely many closed, smooth, oriented surfaces with normal n pointing away from D (these
assumptions allow to use Gauss’s theorem); and 4) the letter S denotes a smooth oriented
surface S, whose boundary ∂S is the union of finitely many C 1 , simple, closed curves oriented
consistently with S (these assumptions allow to use Stokes’s theorem).
1. If F is defined in D, there always exists f such that F = ∇f in D.
TRUE
FALSE.
2. If f is defined in D, there always exists F such that F = ∇f in D.
TRUE
FALSE.
3. If f is defined in D, there always exists F such that f = ∇ · F in D.
TRUE
FALSE.
4. If F is defined on S, and ∇ × F = 0 on S, then the line integral of F along any C 1 ,
simple, closed curve in S is always zero.
TRUE
FALSE.
5. If F is defined in R3 , then the line integral of ∇ × F along ∂S is always zero.
TRUE
FALSE.
6. If f is defined in R3 , then the line integral of ∇f along ∂S is always zero.
TRUE
FALSE.
7. If F is constant (not depending on x, y, z), then the flux of F through ∂D is always
zero.
TRUE
FALSE.
8. If F is defined in R3 , and ∇ · F = 1 in D, then the flux of F through ∂D always equals
the surface area of ∂D.
TRUE
FALSE.
9. If F is defined in R3 , then the flux of ∇ × F through ∂D is always zero.
TRUE
FALSE.
10. If f is defined in R3 , then the flux of ∇f through ∂D is always zero.
TRUE
FALSE.
1
Problem 2. Let C be the cone with parametric equation
(r cos t, r sin t, r)
with 0 ≤ r ≤ 3 and 0 ≤ t ≤ 2π.
a) (4 points) For what order of parameters t and r will the cone parametrized with the above
formula be oriented with the normal pointing away from the z axis.
b) (4 points) Introduce an additional parameter 0 ≤ s ≤ 1 to parametrize a solid bounded
by the above cone and the plane z = 3. (Your answer will be a formula X(s, t, r).)
2
Problem 3. Let T be the solid bounded by the paraboloids z = 8 − x2 − y 2 and z = x2 + y 2 .
Set up, BUT DO NOT EVALUATE, an iterated triple integral that represents the volume
of T
a) (5 points) Viewing T as z-simple.
b) (5 points) Viewing T as x-simple.
3
Problem 4. Let
F(x, y, z) =
1 1 1
, ,
x y z
.
a) (4 points) Find a potential for F in the first octant (x > 0, y > 0, z > 0), or show that F
is not conservative in the first octant.
R
b) (4 points) Show that C F · ds = 0 where C is any simple, C 1 curve whose endpoints
belong to the surface S inside the first octant and satisfying the equation xyz = 4.
c) (4 points) Find a potential for F in the second octant (x < 0, y > 0, z > 0), or show that
F is not conservative in the second octant.
4
Problem 5. (10 points) Let S be the surface given by the parametrization
2
X(t, r) = r cos t, r sin t, er
for 0 ≤ r ≤ 2 and 0 ≤ t ≤ 2π. Find
ZZ
F · dS
S
for F = (2x, −y, −z).
Warning 1: S is not a closed surface!
Warning 2: The parametrization X(t, r) defines an orientation of S.
5
2
2
Problem 6. Let C1 be the ellipse x25 + y9 = 1 and C2 be the unit circle x2 + y 2 = 1, both
oriented counter-clockwise. Let F = yi + 2xj.
H
H
1. (5 points) Show that F · n ds = F · n ds, where n is the outer unit normal (in the
C1
C2
plane!) of the respective curves.
H
H
2. (5 points) Show that F · T ds 6= F · T ds, where T is the unit tangent vector to
C1
C2
the respective curves.
6
Problem 7. (10 points)
Find the area of the region in the first quadrant of R2 that is bounded by the curves:
xy = a2 ,
xy = 2a2 ,
y = x,
7
and y = 2x.
Problem 8. (10 points) Let C be the circle in the plane 2x + y − z = 0, centered at (1, 1, 3),
with radius 3. Let n be the normal vector to the plane 2x + y − z = 0 pointing up. Suppose
C is oriented so that
H T × n points inside the disk in the plane 2x + y − z = 0 bounded by
the circle C. Find F · Tds, where F(x, y, z) = (z, x, y).
C
Hint: Use Stokes’s theorem.
8
Problem 9. (10 points) Let S be the surface defined as the graph of a function g(x, y) ≥ 0
of class C 1 , over the base domain D in the xy−plane, and oriented such that the normal
points up. Let F = (0,
RR0, 1). Prove that, regardless of the choice of g satisfying the above
assumptions, the flux S F · dS is always the same, and equals the area of D.
9