Math 13 — Spring 2020 — Final Exam
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Instructions:
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This is a closed book, closed notes exam. Use of a calculator is not permitted.
You must justify your answers to receive credit. For partial credit, please show your work clearly.
In problems where an answer is required, please circle your final answer.
You have four hours to complete the exam. Please scan this cover page with your submission.
Good luck!
Honor Principle: The Honor Principle requires that you neither give nor receive any aid on this exam. No
books, notes, online sources, or other individuals may be consulted.
For grader use only:
Problem
1
2
3
4
5
6
7
8
9
10
11
12
Total
Points
Score
5
10
10
20
10
10
10+10
10
10
10
5+5+5
10
140
1
2
(1) Express the vector v = (5, 1, 4) as the sum of a vector parallel to w = (1, 2, −1) and a vector
orthogonal to w.
3
(2) Evaluate
Z
0
π
Z
y
π
sin x
dx dy.
x
4
(3) Find the volume of the solid E in the upper half-space lying above the upper sheet of the hyperboloid
given by −x2 − y 2 + z 2 = 1 and below the plane given by z = 2.
5
2
(4) Let D ⊆ R
bounded by the lines x − y = 0, x − y = 2, x + y = 0, and x + y = 3.
RR be the region
2
2
Compute D (x + y)ex −y .
6
BLANK PAGE FOR WORK
7
(5) Consider the vector field F on R2 defined by F(x, y) = (y, x + y) and the curve
α : [0, 1] → R2
R
t(t−1)
−1 2020
given by α(t) = (te
, 2 + (1 − t) tan (t )). Evaluate the line integral α F · ds.
8
(6) Let S be the surface in R3 defined by x2 + y 2 = log z and 1 ≤ z ≤ 2, oriented upwards (i.e., the
unit normal
RR vector field used to orient S at the point (0, 0, 1) has positive z-component). Determine
the flux S F · dS of the vector field F(x, y, z) = (−y, x, z) through the surface S.
9
(7) A regular hexagon centered at the origin in R2 has a pair of opposite
vertices at the points (1, 0) and
√
3
1
(−1, 0) and the remaining four vertices at the points (± 2 , ± 2 ). Let C be the piecewise smooth
path consisting of the sequence of five successive
edges of the hexagon traversed in order counter√
3
1
clockwise,
beginning at the vertex p = (− 2 , 2 ) in the second quadrant and ending at the vertex
√
3
1
q = ( 2 , 2 ) in the first quadrant.
R
(a) For the vector field F(x, y) = (y 2 , 2xy), calculate the line integral C F · ds.
10
(b) Consider the same regular hexagon
R as in problem (7a), and let C be the same piecewise smooth
path. Evaluate the line integral C F · ds, where F(x, y) = (−y, x). [Hint: Think before you
begin computing and save yourself a lot of trouble.]
11
R
−1
(8) Evaluate the line integral C cos(x sin x) dx + (2x + etan y ) dy, where C is the unit circle in R2 ,
oriented counterclockwise.
12
(9) Let F be a C 1 vector field on R3 and let f be a C 1 function on R3 . For a solid region E ⊆ R3 with
C 1 boundary and outward unit normal vector field N on ∂E, show that
ZZZ
ZZ
ZZZ
(0.1)
(∇f ) · F =
(f F · N) dS −
f (div F).
E
∂E
E
13
(10) The Laplacian ∆f of a C 2 function f : R2 → R is defined as ∆f = div(∇f ). Let D ⊆ R2 be a
region in the plane to which Green’s Theorem applies. Prove that
Z
Z
∆f =
(∇n f ) ds,
D
∂D
where ∇v f denotes the directional derivative of f in the direction of a vector v and n is the outward
unit normal vector field on the boundary ∂D.
14
(11) (a) Is the vector field F(x, y) = (2xy, x2 + y 2 ) on R2 the gradient of a C 1 function f : R2 → R?
If so, find the function f .
(b) Is the vector field F(x, y) = (−2xy, x2 + y 2 ) on R2 the gradient of a C 1 function f : R2 → R?
15
(c) Let D = {(x, y) ∈ R2 : x > 0} be the right half-plane in R2 . Consider the vector field
−y
x
F(x, y) =
,
x2 + y 2 x2 + y 2
on D. Is F the gradient of a C 1 function f : D → R?
16
(12) Let v ∈ R3 be a fixed vector.
Show that for a simple closed curve C that bounds an orientable
R
surface S, the line integral C v · ds is zero.
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