Calculus III Homework
Bobby Hanson
i
ii
1
Introduction
The following collection of problems is the entirety of the homework for the semester. I will periodically
(weekly) collect portions of this homework to be graded. I will not accept late homework under any circumstances. However, I will grade the homework as if there were 10% fewer points possible. Essentially, this
means that you could miss one assignment, and still possibly score 100% on the homework. Equivalently,
you can score more than 100% if you miss no assignments. I will not be grading on a curve. This way, when
your classmates score high on the homework, you can congratulate them and feel good for them because
it does not lower your grade. I strongly encourage you to not simply “keep up” with the homework, but
rather, try to stay ahead.
You may work with others (including fellow students, tutors, friends, family, pets, etc.) on any and all
homework problems, but each student should submit their own work. Futhermore, I recommend (but don’t
require) that you draw a star next to any problems that you did without outside help. This is intended to
show you how much you are relying on others. You may use a table of integrals where appropriate, and I
encourage you to use a computer on several of the problems (see below). In fact, you may use a computer
and/or calculator on any problem with the following guideline: you should treat your computer as if it were
your Calculus I/II lackey; that is, you may ask it to perform any task that you might expect a typical
Calculus I/II student to perform. For example, if you are computing a multiple integral, you may ask the
computer to perform any of the single iterations, but not the entire integral itself. The only exception to
this guideline is visualization: you may ask your computer to draw anything you want. Note that you will
not be allowed any type of electronic device for the exams, so you should not become too heavily dependent
on technology in your homework.
That said, I do recommend that you use technology to aid your Calculus education. I have indicated with
a those problems which I feel are deserving of computer use, especially for visualization (the maple leaf
symbol was chosen because I recommend the Maple software; it is available on the department computers
and is easy to learn quickly). I have also used the symbol ´ to indicate that a particular problem should
be done on graph paper if the graphs are drawn by hand. While computer-drawn graphs are quite accurate,
and usually very clean, hand-drawn graphs still have their place.
2
Functions of Two Variables; 15.1
1. Find a newspaper with a weather map for the United States.
(a) Give a range of daily high temperatures for Arizona, New York, California, and Utah.
(b) Sketch the graphs of the predicted high temperature across the U.S. on a north-south line and an
east-west line through Salt Lake City.
2. Consider the acceleration due to gravity, g, at a height h above the surface of a planet of mass m.
(a) If m is held constant, is g an increasing or decreasing function of h? Why?
(b) If h is held constant, is g an increasing or decreasing function of m? Why?
3. You are planning a road trip and your principal cost will be gasoline.
(a) Make a table showing how the daily fuel cost varies as a function of the price of gasoline (in dollars
per gallon) and the number of gallons you buy each day.
(b) If your car goes 30 miles on each gallon of gasoline, make a table showing how your daily fuel cost
varies as a function of your travel distance and the price of gasoline.
4. The table below shows the heat index as a function of temperature and humidity (with temperature
along the top in ◦ F and % humidity down the left side). The heat index is a temperature which tells
you how hot it feels as a result of the combination of the two. Heat exhaustion is likely to occur when
the heat index reaches 105.
0
10
20
30
40
50
60
70
64
65
66
67
68
69
70
75
69
70
72
73
74
75
76
80
73
75
77
78
79
81
82
85
78
80
82
84
86
88
90
90
83
85
87
90
93
96
100
95
87
90
93
96
101
107
114
100
91
95
99
104
110
120
132
105
95
100
105
113
123
135
149
110
99
105
112
123
137
150
115
103
111
120
135
151
(a) If the temperature is 80◦ F and the humidity is 50%, how hot does it feel?
(b) At what humidity does 90◦ F feel like 90◦ F?
(c) Make a table showing the approximate temperature at which heat exhaustion becomes a danger,
as a function of humidity.
(d) Explain why the heat index is sometimes above the actual temperature and sometimes below it.
3
5. This problem concerns a vibrating guitar string. Suppose you pluck a guitar string and watch it vibrate.
If you take snapshots of the guitar string at millisecond intervals, you might get something that looks
like the figure below.
x
0
π
We can analyze the motion of the guitar string using a function of two variables. Think of the guitar
string stretched tight along the x-axis from x = 0 to x = π. Each point on the string has an x value
0 ≤ x ≤ π. As the string vibrates, each point on the string moves back and forth on either side of the
x-axis. The ends of the string at x = 0 and x = π remain stationary, while the point at the middle of
the string moves the most. Let f (x, t) be the displacement at time t of the point on the string located
x units from the left end. Then a possible formula for f (x, t) is
f (x, t) = cos t sin x,
(a)
0 ≤ x ≤ π,
t in milliseconds.
´ Sketch graphs of f versus x for fixed t values t = 0, π/4, π/2, 3π/4, π. Use your graphs to
explain why f represents a vibrating guitar string.
(b) Explain what the functions f (x, 0) and f (x, 1) represent in terms of the vibrating string.
(c) Explain what the functions f (0, t) and f (1, t) represent in terms of the vibrating string.
(d) Compare with the motion of f (x, t) the motions of the guitar strings whose displacements are
given by the following:
g(x, t) = cos 2t sin x,
h(x, t) = cos t sin 2x.
A Tour of Three-Dimensional Space; 14.1, 14.6
6. Which of the points A = (23, 92, 48), B = (−60, 0, 0), C = (60, 1, −92) is closest to the yz-plane? Which
one lies on the xz-plane? Which one is farthest from the xy-plane?
7. Sketch the graphs of the equations
(a) x = −3
(b) y = 1
(c) z = 2 and y = 4 (This is one graph)
8. Describe the set of points whose distance from the x-axis is 2. Describe the set of points whose distance
from the x-axis equals the distance from the yz-plane.
9. Which two of the three points (1, 2, 3), (3, 2, 1), and (1, 1, 0) are closest to each other?
10. Given the sphere
(x − 1)2 + (y + 3)2 + (z − 2)2 = 4,
(a) Find the equations of the circles (if any) where the sphere intersects each coordinate plane.
(b) Find the points (if any) where the sphere intersects each coordinate axis.
4
Graphs of Functions of Two Variables; 14.6, 15.1
11. ´ The surface in the figure below is the graph of the function z = f (x, y) for positive x and y.
(a) Suppose y is fixed and positive. Does z increase or decrease as x increases? Sketch a graph of z
against x.
(b) Suppose x is fixed and positive. Does z increase or decrease as y increases? Sketch a graph of z
against y.
12.
Match the following functions with their graphs below.
(a) z =
1
x2 + y 2
(b) z = −e−x
2
−y 2
(c) z = x + 2y + 3
(d) z = −y 2
(e) z = x3 − sin y
(A)
(C)
(B)
(E)
(D)
5
13. You like pizza and you like coke. Which of the graphs below represents your happiness as a function
of how many pizzas and how much coke you have if
(a) There is no such thing as too many pizzas and too much coke?
(b) There is such a thing as too many pizzas or too much coke?
(c) There is such a thing as too many pizzas, but no such thing as too much coke?
(A)
(C)
(B)
(D)
14. ´ Imagine a single wave traveling along a canal. Suppose x is the distance from the middle of the
canal, t is the time, and z is the height of the water above equilibrium level. The graph of z as a
function of x and t is shown below.
(a) Draw the profile of the wave for t = −1, 0, 1, 2. (Show the x-axis to the right and the z-axis
vertically.)
(b) Is the wave traveling the the direction of increasing or decreasing x?
(c) Sketch a surface representing a wave traveling in the opposite direction.
15. Describe the cross-sections with t fixed and the cross-sections with x fixed of the vibrating guitar string
function
f (x, t) = cos t sin x,
0 ≤ x ≤ π,
from Homework 1. Explain the relation of these cross-sections to the graph of f .
6
Contour Diagrams; 15.1
16. For the following functions, sketch a contour diagram with at least four labeled contours. Describe in
words the contours and how they are spaced.
(a) f (x, y) = x + y
(b) g(x, y) = 2x − y
(c) h(x, y) = xy
(d) k(x, y) = x2 + y 2
17. The figure below shows a contour map of a hill with two paths, A and B.
300
200
100
A
B
(a) On which path, A or B, will you have to climb more steeply?
(b) On which path, A or B, will you probably have a better view of the surrounding countryside?
(Assuming trees do not block your view.)
(c) Near which path is there more likely to be a river?
18. For each of the surfaces below, draw a possible contour diagram, marked with reasonable z values.
(Note: there are many possible answers.)
(a)
(b)
(c)
7
19. Identify the contour diagrams and the surfaces below corresponding to the following equations. Assume
that each contour diagram is drawn in a square window.
(a) z = sin x
(b) z = xy
(c) z = e−(x
2
+y 2 )
(d) z = 1 − 2x − y
(e) z = x2 + 4y 2
(I)
(II)
(IV)
(V)
(F)
(J)
(G)
(K)
(III)
(H)
8
Linear Functions; 15.1
20. Find the equation of the linear function z = c+mx+ny whose graph contains the points (0, 0, 0), (0, 2, −1),
and (−3, 0, −4).
21. Let f be the linear function f (x, y) = c + mx + ny, where c, m, n are constants and m, n 6= 0.
(a) Show that all the contours of f are lines of slope −m/n.
(b) Show that
f (x + n, y − m) = f (x, y),
for all x and y.
(c) Explain the relation between parts (a) and (b).
22. Find the equation of the plane that passes through the points (1, 1, 3), (−1, 2, 2), and (0, 3, 3). [Hint:
Write the general form for a linear function, substitute the given points into it, and solve for the
coefficients.]
23. Give an example of a nonlinear function f (x, y) such that all the cross-sections with x fixed and all
the sections with y fixed are straight lines.
Functions of More Than Two Variables; 15.1
24. Hot water is entering a rectangular swimming pool at the surface of the pool in one corner. Sketch
a possible contour diagram for the temperature of the pool at the surface and three feet below the
surface.
25. Match the following functions with the level surfaces below.
(a) f (x, y, z) = y 2 + z 2 .
(b) f (x, y, z) = x2 + z 2 .
(II)
(I)
26. The figure below shows contour diagrams of temperature in degrees Fahrenheit in a room at three
different times. Describe in as much detail as you can the heat flow in the room. What could be
causing this?
60
65
65
70
70
70
75
75
t =1
80
t =2
t =3
9
27.
Use the catalog of surfaces on pp. 620-621 to identify the surfaces below.
(a) −x2 − y 2 + z 2 = 1
(b) x2 − y 2 − z 2 = 1
(c) −x2 + y 2 − z 2 = 0
(d) x2 + z 2 = 1
(e) x + y = 1
(f) (x − 1)2 + y 2 + z 2 = 1
28. Describe the surface x2 + y 2 = (2 + sin z)2 . In general, if f (z) > 0 for all z, describe the surface
x2 + y 2 = (f (z))2 . [Hint: Think “Pottery”.]
Displacement Vectors; 13.2, 13.3, 13.4, 14.2
29. ´ Draw two vectors ~v and w
~ in the plane at some angle to one another and with different magnitudes.
On this drawing, draw the following vectors:
(a) ~v + w
~
(b) ~v − w
~
(c) 2~v
(d) 2~v + w
~
(e) ~v − 2w
~
30. ´ A cat is sitting on the ground at the point (1, 4, 0) watching a squirrel at the top of a tree. The tree
is one unit high and its base is at the point (2, 4, 0). Find the given displacement vectors below:
(a) From the origin to the cat.
(b) From the bottom of the tree to the squirrel.
(c) From the bottom of the tree to the cat.
(d) From the cat to the squirrel.
31. Which of the following vectors are parallel?
b
~u = 2bi + 4bj − 2k
b
p~ = bi + bj + k
b
~v = bi − bj + 3k
b
~q = 4bi − 4bj + 12k
32. Perform the following operations on the given vectors:
b
~v = 2bj + k
(a) k~zk
w
~ = bi + 6bj
~y = 4bi − 7bj
b
w
~ = −bi − 2bj + k
b
~r = bi − bj + k
~x = −2bi + 9bj
b
~z = bi − 3bj − k
(b) ~v + ~z
(c) 2w
~ + ~x
(d) k~yk
(e) k~y − ~xk
b but has length 2.
33. Find a vector that points in the same direction as bi − bj + 2k
10
Vectors in General; 13.2, 13.3, 13.4, 14.2
34. Say whether the given quantity is a vector or a scalar.
(a)
(b)
(c)
(d)
The
The
The
The
distance from Seattle to St. Louis.
population of the U.S.
magnetic field at a point on the earth’s surface.
temperature at a point on the earth’s surface.
35. ´ Below is a map of the Laguna Seca race track with all of the turns numbered (from 1 to 11). A
car drives counterclockwise around the course, slowing down at the curves and speeding up along the
straight portions of the track. Sketch velocity vectors at each of the numbered curves. A second car
drives clockwise around the track at constant speed. At what point on the track does the car have the
longest acceleration vector, and in roughly what direction is it pointing? Remember that acceleration
is the rate of change of velocity. (There is a page at the end of the homework packet for this problem).
36. ´ Use the geometric definition of addition, subtraction, and scalar multiplication to explain each of
the properties below:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
~u + ~v = ~v + ~u
(α + β)~u = α~u + β~u
α(~u + ~v ) = α~u + α~v
(~u + ~v ) + w
~ = ~u + (~v + w)
~
α(β~u) = (αβ)~u
~v + ~0 = ~v
1 · ~v = ~v
The Dot Product; 13.3, 14.2
b and ~b = −3bi + 5bj + 4k.
b Find ~a · ~b.
37. Let ~a = 2bj + k
b and bi − bj − k.
b
38. Compute the angle between the vectors bi + bj + k
39. Find a normal vector to the plane π(x − 1) = (1 − π)(y − z) + π.
40. Show that the vectors (~b · ~c)~a − (~a · ~c)~b and ~c are perpendicular.
11
The Cross Product; 14.3
b × bj.
41. Find k
42. Does there exist any vector ~u such that ~u × ~u = ~u · ~u? Explain.
43. Find ~a × ~b.
b and ~b = bi + bj.
(a) ~a = bi + k
b and ~b = bi + bj − k.
b
(b) ~a = bi + bj + k
b
(c) ~a = −bi and ~b = bj + k.
b and ~b = bi + 2bj − k.
b
(d) ~a = 2bi − 3bj + k
44. ´ Suppose ~a and ~b are vectors in the xy-plane such that ~a = a1bi+a2bj and ~b = b1bi+b2bj with 0 < a1 < a2
and 0 < b1 < b2 .
(a) Sketch ~a and ~b and the vector ~c = −a2bi + a1bj. Shade the parallelogram formed by ~a and ~b.
(b) What is the relation between ~a and ~c? [Hint: Find ~c · ~a and ~c · ~c.]
(c) Find ~c · ~b.
(d) Explain why ~c · ~b gives the area of the parallelogram formed by ~a and ~b.
b
(e) Verify that in this case ~a × ~b = (a1 b2 − a2 b1 )k.
45. ´ If ~v and w
~ are nonzero vectors, use the geometric definition of the cross product to explain why
(λ~v ) × w
~ = λ(~v × w)
~ = ~v × (λw).
~
Consider the cases λ > 0, λ = 0, and λ < 0 separately.
The Partial Derivative; 15.2
46. The monthly mortgage payment in dollars, P , for a house is a function of three variables
P = f (A, r, N )
where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before
the mortgage is paid off.
(a) Suppose f (92000, 14, 30) = 1090.08. What does this tell you in financial terms?
∂P
(b) Suppose
(92000, 14, 30) = 72.82. What is the financial significance of the number 72.82?
∂r
∂P
to be positive or negative? Why?
(c) Would you expect
∂A
∂P
(d) Would you expect
to be positive or negative? Why?
∂N
47. A drug is injected into a patient’s blood vessel. The function c = f (x, t) represents the concentration
of the drug at a distance x in the direction of the blood flow measured from the point of injection and
at time t since the injection. What are the units of the following partial derivatives? What are their
practical interpretations? What do you expect their signs to be?
∂c
∂x
∂c
(b)
∂t
(a)
12
48. Estimate zx (1, 0), zx (0, 1), zy (0, 1) from the figure below:
y
1.2
1.0
0.8
0.6
0.4
z =1
z =2
1
1.5
z =3
0.2
0.5
2
x
49. Below is the table from Problem 4, showing the heat index, I, as a function f (h, T ) of the humidity, h,
and the temperature, T . (Temperature is along the top in ◦ F and % humidity is down the left side).
0
10
20
30
40
50
60
70
64
65
66
67
68
69
70
75
69
70
72
73
74
75
76
80
73
75
77
78
79
81
82
85
78
80
82
84
86
88
90
90
83
85
87
90
93
96
100
95
87
90
93
96
101
107
114
100
91
95
99
104
110
120
132
105
95
100
105
113
123
135
149
110
99
105
112
123
137
150
115
103
111
120
135
151
∂I
∂I
and ∂T
for typical weather conditions in Tucson in summer (h = 10, T = 100). What
(a) Estimate ∂h
do your answers mean in practical terms for the residents of Tucson?
(b) Answer the question in part (a) for Boston in summer (h = 50, T = 80).
50. ´ Suppose that c represents the cardiac output, which is the volume of blood flowing through a person’s
heart, and that s represents the systemic vascular resistance (SVR), which is the resistance to blood
flowing through veins and arteries. Let p be a person’s blood pressure. Then p = f (c, s) is a function
of c and s.
∂p
represent?
∂c
(b) Suppose p = kcs, where k is a positive constant. Sketch the level curves of p. What do they
represent? Label your axes.
(a) What does
(c) For a person with a weak heart, it is desirable to have the heart pumping against less resistance,
while maintaining the same blood pressure. Such a person is given the drug Nitroglycerine to
decrease the SVR and the drug Dopamine to increase the cardiac output. Represent this on
a graph showing level curves (use the same function p = kcs). Put a point A on the graph
representing the person’s state before the drugs are given and a point B for after.
(d) Right after a heart attack, a patient’s cardiac output drops, thereby causing the blood pressure
to drop. A common mistake made by medical residents is to get the patient’s blood pressure back
to normal by using drugs to increase the SVR, rather than by increasing the cardiac output. On
a graph of the level curves of p, put a point D representing the patient before the heart attack,
a point E representing the patient right after the heart attack, and a third point F representing
the patient after the resident has given the drugs to increase the SVR.
13
Computing Partial Derivatives Algebraically; 15.2, 15.4
51. Let f (u, v) = u(u2 + v 2 )3/2 .
(a) Use a difference quotient to approximate fu (1, 3) with h = 0.001.
(b) Now evaluate fu (1, 3) exactly. Was the approximation in part (a) reasonable?
52. Find the indicated partial derivatives. Assume the variables are restricted to a domain on which the
function is defined.
(a)
(b)
(c)
(d)
(e)
(f)
1
∂A
if A = (a + b)/h
∂h
2
mv 2
Fv if F =
r
1
∂
2
mv
∂m 2
1
1 2
uE if u = ǫ0 E 2 +
B
2
2µ0
∂F
Gm1 m2
if F =
∂m2
r2
1
∂
v0 t + at2
∂t
2
3x2 y 7 − y 2
15xy − 8
∂m
m0
(h)
if m = q
2
∂v
1 − vc2
(g) zy if z =
(i)
∂α
exp(xβ − 3)
if α =
∂β
2yβ + 5
53. Suppose you know that
fx (x, y)
fy (x, y)
=
=
4x3 y 2 − 3y 4 ,
2x4 y − 12xy 3 .
Can you find a function f which has these partial derivatives? If so, are there any others?
14
Local Linearity and the Differential; 15.7
54. Find the equation of the tangent plane to z = yex/y at the point (1, 1, e).
55. If z = e3x sin y, find dz.
56. Find the differentials to the following functions:
(a) f (x, y) = cos(x2 y).
(b) g(s, t) = s4 + ts2 .
(c) θ(α, β) = e0.1α cos(2πβ + 10α).
57. Find the differentials of the functions in Problem 56 at the points:
(a) (x, y) = (1, 2).
(b) (s, t) = (3, 5).
(c) (α, β) = (π, 2).
58. One mole of ammonia gas is contained in a vessel which is capable of changing its volume (e.g. a
piston). The total energy U (in Joules) of the ammonia is a function of the volume V (in m3 ) of the
container, and the temperature T (in Kelvin) of the gas. The differential dU is given by
dU = 840 dV + 27.32 dT.
(a) How does the energy change if the volume is held constant and the temperature is increased
slightly?
(b) How does the energy change if the temperature is held constant and the volume is increased
slightly?
(c) Find the approximate change in energy if the gas is compressed by 100cm3 and heated by 2 K.
59. The coefficient of thermal expansion of a liquid, β, relates the change in its volume V (in m3 ) to an
increase in its temperature T (in ◦ C):
dV = βV dT
(a) Let ρ be the density (in kg/m3 ) of 1 kg of water as a function of temperature. Write an expression
for dρ in terms of ρ and dT .
(b) The graph below shows the density of water as a function of temperature. Use it to estimate β
when T = 20◦ C and when T = 80◦ C.
ρ
1000
990
980
970
960
20 40 60 80 100
T
15
Directional Derivatives; 15.5
60. Use a difference quotient to estimate the rate of change of f (x, y) = xey at the point (1,1) as you move
in the direction of the vector bi + 2bj.
61.
´ Draw a contour diagram for the function z = f (x, y) = x2 + 4y 2 on the window −2 ≤ x ≤ 2 and
−2 ≤ y ≤ 2 with the contours z = 0, z = 1, z = 2, z = 3, z = 4. Try to be as accurate as possible. Use
your contour diagram to estimate each of the following directional derivatives.
(a) fbi (0, 1).
(b) fbj (0, 1).
√
(c) fu~ (1, 1), where ~u = (bi − bj)/ 2.
√
(d) f~v (1, 1), where ~v = (bi + bj)/ 2.
62. Evaluate analytically the directional derivatives from Problem 61.
63. Find the directional derivative of f (x, y, z) = x2 y + y 2 z at the point (1, 2, 3) in the direction of the
following vectors (note that they are not unit vectors):
b
(a) ~u = −bj + k.
b
(b) ~v = 2bi + 3bj + 5k.
The Gradient; 15.5
64. Suppose that f is differentiable at (a, b). Then there is always a direction in which the rate of change
of f at (a, b) is 0. True or false? Explain your answer.
65. Compute the gradient of the given functions.
(a) f (x, y) = x3 + y 2
(b) g(u, v) = uv
(c) h(x, t) = eαt sin
πx L
66. Find grad z at the specified point.
(a) z = sin(x/y), at (π, 1)
xey
, at (1, 2)
(b) z = 2
x + y2
67. Let f (x, y) =
vectors
x+y
1+y 2
(a) ~v = 3bi − 2bj,
(b) ~u = −bi + 4bj.
and let A = (2, 3). Find the directional derivative at A in the direction of the
(c) What is the direction of greatest increase at A?
16
68.
The temperature at any point in the plane is given by the function
T (x, y) =
100
.
x2 + y 2 + 1
(a) Where on the plane is it hottest? What is the temperature at that point?
(b) Find the direction of the greatest increase in the temperature at the point (3, 2). What is the
magnitude of that greatest increase?
(c) Find the direction of the greatest decrease in temperature at the point (3, 2).
(d) Is the direction of the vector you found in part (b) towards the point you found in part (a)?
(e) Find a direction at the point (3,2) in which the temperature does not increase or decrease (See
Problem 64).
(f) What shape are the level curves of T ?
69.
√
´ Sketch the level curves of the function z = f (x, y) = xy on the window 0 ≤ x ≤ 5 and 0 ≤ y ≤ 5.
You should have the contours z = 1, z = 2, z = 3, and z = 4. At the points (1, 1) and (1, 4) on
your sketch, draw a vector representing ∇f (no computations required). Explain how you decided the
approximate length and direction of each vector.
70. Consider S to be the surface represented by the equation F = 0, where
F (x, y, z) = x2 z 2 − y.
(a) Find all points on S where a normal vector is parallel to the xy-plane.
(b) Find the tangent plane to S at the points (0, 0, 1) and (1, 1, 1).
(c) Find the unit vectors ~u1 and ~u2 pointing in the direction of maximum increase of F at the points
(0, 0, 1) and (1, 1, 1) respectively.
The Chain Rule; 15.6
71. Find
dz
dt
using the chain rule.
(a) z = xy 2 ,
x = e−t ,
y
(b) z = xe ,
72. Find
∂z
∂u
and
x = 2t,
y = sin t
y = 1 − t2
∂z
∂v
(a) z = xe−y ,
x = u sin v,
(b) z = sin(x/y),
x = ln u,
, y = v cos u
y=v
73. This problem is an exercise in changing between Cartesian and polar coordinates (this change of
coordinates is routine in the sciences). Suppose the quantity z can be expressed either as a function of
Cartesian coordinates (x, y) or as a function of polar coordinates (r, θ), so that z = f (x, y) = g(r, θ).
(a) use the chain rule to find
∂z
∂r
and
∂z
∂θ
in terms of
∂z
∂x
(b) Solve the equations you have just written down for
and
∂z
∂x
∂z
∂y .
and
∂z
∂y .
(c) Show that the expressions you get in part (b) are the same as you would get by using the chain
∂z
∂z
∂z
and ∂z
rule to find ∂x
∂y in terms of ∂r and ∂θ .
(d) Show that
∂z
∂x
2
+
∂z
∂y
2
=
∂z
∂r
2
1
+ 2
r
∂z
∂θ
2
.
74. Suppose w = f (x, y, z) and that x, y, z are functions of u, v. Draw a tree diagram representing the
∂w
relation between the variables and use it to derive the chain rule formula for ∂w
∂u and ∂v .
17
Second-Order Partial Derivatives; 15.2
75. Calculate all four second-order partial derivatives and show that zxy = zyx .
x
(a) z = sin
y
(b) z = xey
(c) z = xy
(d) z = ln(xy)
76. If z = f (x) + yg(x) what can you say about zyy ? Why?
77. If zxy = 4y, what can you say about the value of
(a) zyx ?
(b) zxyx ?
(c) zxyy ?
Partial Differential Equations
78.
´ Sketch the graph of u(x, t) = 1 − (x − 2t)2 for t = 0, 1, 2. Do you see a traveling wave? What is
its speed?
79. We will model the spread of an epidemic through a region. Let I(x, y, t) represent the density of sick
people per unit area at the point (x, y) in the plane at time t. A good model suggests that I should
satisfy the diffusion equation:
2
∂ I
∂I
∂2I
+ 2
=k
∂t
∂x2
∂y
where k is a constant. Suppose we know that for some particular epidemic
I = eax+by+ct .
What can you say about the relationship between a, b, and c?
80. For what values of the constants a and b will the function u = (x + y)eax+by satisfy the partial
differential equation
∂2u
∂u ∂u
−
−
+ u = 0?
∂x∂y ∂x ∂y
81. Show that the functions below satisfy Laplace’s equation, Fxx + Fyy = 0.
(a) F (x, y) = ex sin y + ey sin x.
(b) F (x, y) = eax+by , if a2 + b2 = 0 (note that in this case a and b are complex numbers).
82. Suppose that f is any differentiable function of one variable. Define V , a function of two variables by
V (x, t) = f (x + ct).
Show that V satisfies the partial differential equation
∂V
∂V
=c
.
∂t
∂x
18
83. Consider the wave equation:
2
∂2y
2∂ y
=
a
∂t2
∂x2
where a is a constant, and y is the displacement of the wave at any point x and at any time t (so y is
a function of x and t). Write the boundary conditions for a vibrating string of length L for which:
(a) The ends (at x = 0 and x = L) are fixed.
(b) The initial shape is given by f (x).
(c) The initial velocity distribution is given by g(x).
84. The gravitational potential, V , at any point (x, y, z) outside a spherically symmetric mass m located
at the point (0, 0, 0), is defined as V = − Gm
r , where r is the distance from (x, y, z) to the mass (origin)
and G is a constant. Show that V satisfies Laplace’s equation:
∂ 2V
∂2V
∂2V
+
+
= 0,
∂x2
∂y 2
∂z 2
for all points outside the mass.
85. (a) Find the relationship between a and b that must hold for u(x, t) = eat sin(bx) to satisfy the heat
equation ut = uxx .
(b) Suppose you wish to study heat conduction in a 1 meter metal rod, 0 ≤ x ≤ 1, wrapped in
insulation, whose ends are maintained at 0◦ C at all times (for instance because they are stuck
into ice baths). The conditions at the ends of the rod represent a boundary condition on the
possible functions u(x, t) that could describe the temperature in the rod. The boundary condition
must hold in addition to the PDE ut = uxx . State the boundary condition as a pair of equations.
(c) Determine all possible values of a and b such that u(x, t) = eat sin(bx) satisfies both the PDE and
the boundary condition of part (b).
86. The temperature T of a metal plate can be described by a function T = u(x, y, t) of three variables:
the two space variables x and y, and the time variable t. In many situations this function will satisfy
the two-dimensional heat equation:
ut = A(uxx + uyy )
where A is a positive constant. Find conditions on a, b, and c such that
u(x, y, t) = e−at sin(bx) sin(cy)
satisfies this equation.
Local and Global Extrema; 15.8
87.
For each of the following functions, find the global maximum and minimum over the square −1 ≤ x ≤ 1,
−1 ≤ y ≤ 1, and say whether it occurs on the boundary of the square. (Hint: Consider the graph of
the function).
(a) z = x2 + y 2
(b) z = x2 − y 2
(c) z = −x2 − y 2
88. Each of the following functions has a critical point at (0, 0). What sort of critical point is it? See if
you can do this without using the Second Derivative Test.
(a) f (x, y) = x4 + y 4
(b) g(x, y) = x6 + y 5
(c) h(x, y) = cos x cos y
19
89. Find the local maxima, local minima, and saddle points of the function f (x, y) = x2 + y 3 − 3xy.
90. Suppose that for some function f (x, y) at the point (a, b), we have fx = fy = 0, fxx > 0, fyy > 0,
fxy = 0.
(a) What can you conclude about the behavior of the function near the point (a, b)?
(b) Sketch a possible contour diagram.
91. Suppose that for some function f (x, y) at the point (a, b), we have fx = fy = 0, fxx > 0, fyy = 0,
fxy > 0.
(a) What can you conclude about the behavior of the function near the point (a, b)?
(b) Sketch a possible contour diagram.
92.
(See http://www.math.hmc.edu/faculty/gu/curves and surfaces/surfaces/monkey.html)
The hyperbolic paraboloid f (x, y) = xy is often called a saddle surface, because a person could sit on
it comfortably. A monkey, however, would run into trouble because he would have nowhere to put
his tail! The below surface is called a monkey saddle, because it has a convenient dip in the back to
accommodate the monkey’s tail. The monkey saddle is the graph of the function g(x, y) = x3 − 3xy 2 .
In this problem we will compare these two functions.
(a) Verify that (0, 0) is indeed a saddle point for both the hyperbolic paraboloid (human saddle) and
the monkey saddle.
(b) Now we make new functions:
F (x, y) = f (x, y) + 3x = xy + 3x
G(x, y) = g(x, y) + 3x = x3 − 3xy 2 + 3x.
This has the result of tilting the graph slightly, without changing its shape. Explain, in words,
why this is the case.
(c) Find the critical points of the new functions F (x, y) and G(x, y) (the tilted saddles).
(d) What kind of critical points are they?
(e) Summarize, in words, what happened to the saddle points of f (x, y) and g(x, y) after we tilted
their graphs.
(f) Use Maple to graph the contour diagrams of these four functions. Find the critcal points on your
graphs.
20
Unconstrained Optimization; 15.8
93. Find the shortest distance from the origin to the plane x + 2y + 3z = 12.
94. Find the minimum distance between the point (2, 1, 0) and the cone z 2 = x2 + y 2 .
95. Find the maximum and minimum values of z = x2 − y 2 on the closed triangle with vertices (0, 0),
(1, 2), and (2, −2).
96.
Assume that two products are manufactured in quantities q1 and q2 and sold at prices p1 and p2
respectively, and that the cost of producing them is given by
C = 2q12 + 2q22 + 10.
(a) Find the maximum profit that can be made, assuming the prices are fixed.
(b) Find the rate of change of that maximum profit as p1 increases.
97.
A mountain climber reaches the peak of a mountain late in the day. After taking photographs of the
view from the top, the weather deteriorates, and she has to descend the mountain as quickly as possible
to seek shelter at a lower altitude. The altitude of the mountain (in feet) is given approximately by
h(x, y) = 14, 000 −
1
5x2 + 4xy + 2y 2 ,
24000
where x, y are the horizontal coordinates on the earth (in feet), with the mountain summit located
above the origin. In fifteen minutes, the climber can reach any point (x, y) on a circle of radius 2000
feet. In which direction should she travel in order to descend as far as possible?
21
Constrained Optimization – Lagrange’s Method; 15.9
98. The figure below is a topographical map showing three mountains and a couple of passes. We have
decided to hike the trail indicated by the dashed line. (There is a page at the end of the packet for
this problem)
(a) On the map, place an X at each critical point along the trail; that is, at each point where the
gradient of the landscape is perpendicular to the trail.
(b) Which of these critical points are maxima, minima, or saddle points?
(c) What are the highest and lowest points we reach along the trail?
8600
8600
8800
+ 10320
10000
9000
+ 9772
9600
9800
9400
9200
9600
9400
9000
9000
9200
+ 9731
9600
9000
9400
8800
8600
8400
9200
8200
8000
7800
99.
Use Lagrange multipliers to find the maximum and minimum values of f (x, y) subject to the given
constraints.
(a) f (x, y) = x + y,
(b) f (x, y) = x2 + y,
(c) f (x, y) = xy,
x2 + y 2 = 1
x2 − y 2 = 1
4x2 + y 2 = 8
100. Design a closed cylindrical container which holds 100cm3 and has the minimal possible surface area.
What should its dimensions be?
101. Find the shortest distance from the origin to the surface x2 y − z 2 + 9 = 0.
22
The Definite Integral of a Function of Two Variables; 16.1, 16.3
√
102. ´ Let R be the rectangle with vertices (0, 0), (4, 0), (4, 4), and (0, 4) and let f (x, y) = xy.
Z
f dA without subdividing R.
(a) Find reasonable upper and lower bounds for
R
Z
(b) Estimate
f dA by partitioning R into four subrectangles and evaluating f at its maximum and
R
minimum values on each subrectangle.
103. A biologist studying insect populations measures the population density of flies and mosquitos at various points in a rectangular study region. The graphs of the two population densities for the region are
shown below. Assuming that the units along the corresponding axes are the same in the two graphs,
are there more flies or more mosquitos in the region? Why?
(a)
(b)
Iterated Integrals; 16.2
104. Evaluate the given integrals:
Z
√
x + y dA, where R is the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.
(a)
ZR
(5x2 + 1) sin 3y dA, where R is the rectangle −1 ≤ x ≤ 1, 0 ≤ y ≤ π3 .
(b)
ZR
(2x + 3y)2 dA, where R is the triangle with vertices at (−1, 0), (0, 1) and (1, 0).
(c)
R
23
105. ´ For each of the regions R below, set up
R
R
f dA as an iterated integral.
y
y
1
2
-1
1
(a)
1
2
3
4
x
(b)
2
1
3
x
-2
y
y
2
3
2
1
1
(c)
1
2
3
4
x
(d)
1
2
3
x
106. ´ For each of the following integrals, sketch the region of integration and evaluate the integral
Z 3Z 4
ex+y dy dx
(a)
1
(b)
Z
0
2
0
(c)
Z
1
(d)
Z
(e)
Z
1
Z
x
2
ex dy dx
0
5
Z
2x
sin(x) dy dx
x
4Z y
√
0
−2
Z
x2 y 3 dx dy
y
4
√
− 9−x2
2xy dy dx
107. ´ Consider the integral
Z
0
4
Z
−(y−4)/2
g(x, y) dx dy.
0
(a) Sketch the region over which the integration is being performed.
(b) Write the integral with the order of integration reversed.
108. Evaluate the following integrals by reversing the order of integration.
Z 1Z 1 p
2 + x3 dx dy
(a)
√
y
0
(b)
Z
(c)
Z
1
0
(d)
1
2
ex dx dy
y
3
Z
9
y sin(x2 ) dx dy
y2
0
Z
Z
0
−4
Z
0
2x+8
f (x, y) dy dx +
Z
0
−4
Z
2x+8
f (x, y) dy dx
0
109. ´ Find the average distance to the x-axis for points in the region bounded by the x-axis and the graph
of y = x − x2 .
24
110. Evaluate
Z 1Z
(a)
0
(b)
Z
1
sin(x2 ) dx dy
y
1
Z
e
ey
0
x
dx dy
ln x
Three-Variable Integrals; 16.7
111. ´ Find the three-variable integral of the given functions over the given regions.
(a) f (x, y, z) = x2 + 5y 2 − z, W is the rectangular box 0 ≤ x ≤ 2, −1 ≤ y ≤ 1, 2 ≤ z ≤ 3
(b) g(x, y, z) = sin x cos(y + z), W is the rectangular box 0 ≤ x ≤ π, 0 ≤ y ≤ π, 0 ≤ z ≤ π
(c) h(x, y, z) = ax + by + cz, W is the rectangular box 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1
112. ´ For the following problems, describe or sketch the region of integration for the triple integrals. If
the limits do not make sense, explain why.
Z 6 Z 3−x/2 Z 6−x−2y
(a)
f (x, y, z) dz dy dx
0
(b)
Z
0
1
0
(c)
Z
Z
1
Z
3
(f)
3
1
(g)
−1
f (x, y, z) dz dy dx
f (x, y, z) dz dy dx
0
Z
√
9−y 2
x+y
Z
2−x
0
1
0
x
0
1
0
Z
Z
Z
0
zZ x
−
1
Z
Z
0
0
(e)
x
0
0
(d)
Z
Z
0
Z
Z
Z
3
√
f (x, y, z) dz dx dy
x2 +y 2
y
f (x, y, z) dz dx dy
0
3
f (x, y, z) dz dy dx
√ 2 2
2 Z
0
√
1−x
2−x −y
f (x, y, z) dz dy dx
0
113. Find the average value of the sum of the squares of three numbers x, y, z where each number is between
0 and 2.
114. ´ Find the volume of the solid formed by the intersections of the cylinders x2 + z 2 = 1 and y 2 + z 2 = 1.
25
Two-Variable Integrals in Polar Coordinates; 16.4
115. ´ For each of the regions R below, set up
Z
f dA as an iterated integral in polar coordinates.
R
y
2
y
2
1
x
-2
-1
0
(a)
1
x
2
1
(b)
y
3
(c)
-3
y
0.5
3 x
-3
x
-0.5
(d)
0.5
-0.5
116. ´ Sketch the regions over which the following integrals are computed.
Z 2π Z 2
(a)
f (r, θ)r dr dθ
0
(b)
Z
π
π/2
(c)
Z
Z
Z
π/3
4
3
(e)
Z
(f)
Z
(g)
0
Z
1
f (r, θ)r dr dθ
3π/2
f (r, θ)r dθ dr
3π/4
π/4
Z
1/ cos θ
f (r, θ)r dr dθ
0
π/2
π/4
Z
f (r, θ)r dr dθ
0
0
Z
1
0
π/6
(d)
1
4
Z
2/ sin θ
f (r, θ)r dr dθ
0
Z
π/2
−π/2
f (r, θ)r dθ dr
2
26
117. ´ Evaluate the integrals below over the region indicated
Z
(a)
sin(x2 + y 2 ) dA where R is the disc of radius 2 centered at the origin.
R
Z
(b)
(x2 − y 2 ) dA where R is the first quadrant region between the circles of radius 1 and radius 2.
R
118. ´ Change the following integrals to polar coordinates and evaluate.
Z 0 Z √1−x2
x dy dx
(a)
√
(b)
−1 − 1−x2
√ Z √
2
4−y 2
Z
0
xy dx dy
y
2
119. ´ A disk of radius 5 cm has density 10 g/cm at its center, has density 0 at its edge, and its density
is a linear function of the distance from the center. Find the mass of the disk.
Integrals in Cylindrical and Spherical Coordinates; 16.8, 14.7
120. ´ Evaluate the indicated integrals in cylindrical coordinates.
Z
x2 + y 2 + z 2 dV where W is the region 0 ≤ r ≤ 4, π/4 ≤ θ ≤ 3π/4, −1 ≤ z ≤ 1.
(a)
W
Z
sin(x2 + y 2 ) dV where W is the solid cylinder with height 4 and with base of radius 1 centered
(b)
W
on the z-axis at z = −1.
121. ´ Evaluate the indicated integrals in spherical coordinates.
Z
1
p
dV where W is the bottom half of the sphere of radius 5 centered at the origin.
(a)
x2 + y 2 + z 2
W
Z
sin φ dV where W is the region 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/4, 1 ≤ ρ ≤ 2.
(b)
W
122. ´ Sketch the region over which the integration is being performed:
Z
0
π/2
Z
π
π/2
Z
1
f (ρ, φ, θ)ρ2 sin φ dρ dφ dθ.
0
123. ´ Evaluate the following integrals:
Z 1 Z √1−x2 Z √1−x2 −z2
1
p
dy dz dz.
(a)
√
√
2
2
2
2
x + y2 + z 2
0
− 1−x
− 1−x −z
Z 1 Z 1 Z √1−x2
1
p
(b)
dy dx dz.
√
2
x + y2
0
−1 − 1−x2
124. ´ Find the volume that remains after a cylindrical hole of radius a is bored through a sphere of radius
R, where 0 < a < R, passing through the center of the sphere along the pole.
27
Motion in Space; 13.4, 14.4
125. ´ Describe the motion of a particle whose position at time t is x = f (t), y = g(t), where the graphs of
f and g are as shown below.
x
1
y
1
t
t
1
2
3
4
2
3
4
1
2
3
4
-1
y
1
(a) -1
x
2
t
1
t
(b)
1
1
2
3
4
-1
126. For the following parameterizations of the motion of a particle, the curve being traced out is a circle.
Describe in words how the circle is traced out, including when and where the particle is moving
clockwise and when and where the particle is moving counterclockwise.
(a) x = cos(t3 − t), y = sin(t3 − t)
(b) x = cos(cos t), y = sin(cos t)
(c) x = cos(ln t), y = sin(ln t)
127. ´ Describe the similarities and differences among the motions in the plane given by the three following
pairs of parametric equations:
(a) x = t, y = t2
(b) x = t2 , y = t4
(c) x = t3 , y = t6
128.
´ In class we saw the curve known as a helix, given by x = cos t, y = sin t, z = t. Imagine a light
shining on the helix from far down each of the axes. Sketch the shadow cast by the helix on each of
the coordinate planes: xy, xz, and yz.
Parameterized Curves; 13.4, 14.4
129. ´ Write a parameterization for each of the following curves in the xy-plane.
(a) A circle of radius 3 centered at the origin and traced out clockwise.
(b) A line through the point (1, 3) and parallel to the vector −2bi − 4bj.
(c) A vertical line through the point (−2, 3).
(d) A circle of radius 5 centered at the point (2, 1) and traced out counterclockwise.
(e) A circle of radius 2 centered at the origin traced out clockwise starting at the point (−2, 0) when
t = 0.
(f) An ellipse centered at the origin and crossing the x-axis at ±5 and the y-axis at ±7. (The axes of
the ellipse are along the coordinate axes). I don’t care whether your ellipse is traced out clockwise
or counterclockwise, but please state which it is.
28
130. ´ If t is allowed to take on all real values, the parametric equations
x = 2 + 3t,
y = 4 + 7t
describe a line in the plane.
(a) What part of the line is obtained by restricting t to nonnegative numbers?
(b) What part of the line is obtained if t is restricted to −1 ≤ t ≤ 0?
(c) How should t be restricted to give the part of the line to the left of the y-axis?
131. Two particles are traveling through space. At time t the first particle is at the point (−1+t, 4−t, −1+2t)
and the second is at (−7 + 2t, −6 + 2t, −1 + t).
(a) Describe the two paths.
(b) Will the two particles collide, if so when and where?
(c) Do the paths of the two particles cross, and if so where?
132. Each of the three circles A, B and C of the figure below can be parameterized by equations of the form
x = a + k cos t,
y = b + k sin t,
0 ≤ t ≤ 2π.
What can you say about the values of a, b and k for each of these circles?
y
10
B
C
x
10
A
-10
133.
´ What curves do the following sets of parametric equations trace out? Find an implicit or explicit
equation for each curve.
(a) x = 2 + cos t, y = 2 − sin t
(b) x = 2 + cos t, y = 2 − cos t
(c) x = 2 + cos t, y = cos2 t
29
Velocity and Acceleration Vectors; 13.5, 14.5
134. (a) Sketch the parameterized curve x = t cos t, y = t sin t for 0 ≤ t ≤ 4π.
(b) Use difference quotients to approximate the velocity vectors ~v (t) for t = 2, 4 and 6.
(c) Compute exactly the velocity vectors ~v (t) for t = 2, 4 and 6, and sketch them onto the graph of
the curve.
135. Each of the following parameterized curves represents the motion of a particle. For each motion, find
the velocity vector ~v (t) for the particle, and the acceleration ~a(t) for the particle. Also, find the speed
of the particle k~v (t)k and any times when the particle comes to a stop.
(a) x = t2 ,
y = t3
(b) x = cos(t2 ),
y = sin(t2 )
(c) x = cos(2t),
y = sin(t)
(d) x = t,
, y = t2 ,
z = t3 .
136. For each of the curves in Problem 135, find the parametric equations for the tangent line at t = 2.
137. Find the length of the following curves and explain why your answer is reasonable.
(a) x = 3 + 5t,
y = 1 + 4t,
t
t
(b) x = cos(e ),
z = 3 − t for 1 ≤ t ≤ 2
y = sin(e ) for 0 ≤ t ≤ 1.
√
b
b
(c) ~r(t) = t cos ti + t sin tbj + 2tk.
138. Consider the motion of a particle given by the parametric equations x = t3 − 3t,
y = t2 − 2t.
(a) Does the particle ever come to a stop? If so, when and where?
(b) Is the particle ever moving straight up and down (parallel to the y-axis)? If so, when and where?
(c) Is the particle ever moving straight horizontally left or right? If so, when and where?
b represents the position of a particle on a helix at time t where z is the
139. Suppose ~r = cos tbi + sin tbj + tk
height of the particle above the ground.
(a) Is the particle ever moving downwards? If so, when and where?
(b) When does the particle reach a point 10 units above the ground?
(c) What is the velocity of the particle at that time?
(d) Suppose the particle leaves the helix and moves along the tangent line the the spiral at this point.
What is the equation of the tangent line?
b the principal normal N,
b
140. For each of the curves below, find the curvature κ, the unit tangent vector T,
b at the time t = t1 given.
and the binormal B
b
(a) ~r(t) = 13 t3bi + 21 t2 k;
t1 = 1.
b
(b) ~r(t) = e−tbi + etbj + 2tk;
t1 = 0.
b in Problem 140b, find the tangential and normal components, aT
141. For the curve ~r(t) = e−tbi + etbj + 2tk,
and aN , of the acceleration vector at any time t.
b =T
b × N.
b Prove that N
b =B
b × T.
b Hint: you only need to
142. Recall that the unit binormal vector is B
use properties of cross-products.
b is perpendicular to T
b by diferentiating T
b ·T
b = 1 with respect to s.
143. Prove that N
b
b =T
b ×N
b has the property that dB is perpendicular to B,
b and
144. Show that the unit binormal vector B
ds
b Hint: for the first part, consider Problem 143; for the second part, use the product
perpendicular to T.
rule for derivatives (it applies to cross-product as well as dot-product!).
30
b
dB
b and, consequently, there must
must be parallel to N
ds
b
dB
b The function τ (s) is call the torsion of the
be a number τ depending on s such that
= −τ (s)N.
ds
b and N.
b
curve and measures the twist of the curve from the plane determined by T
145. Using the results of Problem 144, show that
146. Compute the torsion of the helix given in Problem 139, above.
147. Chewbacca is flying the Millenium Falcon along the curve given by
b
~r(t) = −2e3tbi + 5 cos tbj − 3 sin(2t)k.
If the wookie turns off his ion engines, the Falcon will fly off along a tangent line to ~r(t). He is
almost out of power when he notices that a station on Yavin V is open at the point with coordinates
(1.5, 5, 3.5). Quickly calculating his position, he turns off the thrusters at t = 0. Does he make it to
the Yavin V station? Explain.
b
148. A bee was flying along a helical path so that its position vector at time t was ~r(t) = cos tbi+sin tbj+16tk.
At time t = 12, it had a heart attack and died instantly. Where did it land (i.e. hit the xy-plane)?
Assume that distance is in feet, time is in seconds, and g = 32 feet per second per second.
149. An object moves along the curve y = sin x. Without doing any calculations, decide where aN = 0.
~
150. The angular momentum L(t)
and torque ~τ (t) of a moving particle of mass m and position vector ~r(t)
are
~
L(t)
= m~r(t) × ~v (t), ~τ (t) = m~r(t) × ~a(t).
~ ′ (t) = ~τ (t).
Show that L
100
gives the temperature at the point (x, y) in the plane. Suppose that
1 + x2 + y 2
a ladybug moves along a parabola according to the parametric equations x = t, y = t2 . Find the
instantaneous rate of change in the temperature of the ladybug at time t. See Problem 68
151. Suppose T (x, y) =
152. This problem generalizes the result of Problem 151. Suppose that T (x, y) gives the temperature at any
point (x, y) in the plane and that a ladybug moves in the plane with position vector at time t given by
d
~r(t) = x(t)bi + y(t)bj and velocity vector ~v (t) = dt
~r(t). Use the chain rule to show that
Rate of change in the temperature of the bug at time t = ∇T (x(t), y(t)) · ~v (t).
31
153. A lighthouse L is located on an island in the middle of a lake as shown below. Consider the motion of
the point where the light beam from L hits the shore of the lake.
C
A
B
L
D
E
(a) Suppose the beam rotates counterclockwise about L at constant angular velocity. At which of
A, B, C, D or E is the speed of the point greatest, and at which point smallest?
(b) Repeat part (a), supposing the beam rotates counterclockwise so that it sweeps out equal areas
of the lake in equal times.
(c) What happens if you place the lighthouse at different points in the lake? Can the speed of the
point ever be infinite if the beam rotates at constant angular velocity?
(d) Suppose now that the lake is rectangular instead. What happens to the velocity vector at the
corners? For part (b) show that the speed is constant along each side (possibly a different constant
for each side).
Parameterized Surfaces
154. Describe in words each of the following curves on the surface of the globe:
(a) φ = π/4
(b) θ = π/4
155. Find parametric equations for
b
(a) The plane through (1, 3, 4) and orthogonal to ~n = 2bi + bj − k.
(b) The sphere centered at the origin and having radius 5.
(c) The sphere centered at the point (2, −1, 3) and with radius 5.
(d) The sphere x2 + y 2 + z 2 = R2
(e) The cone x2 + y 2 = z 2 .
156. Adapt the parameterization for the sphere (see Problem 155c) to find a parameterization for the
ellipsoid
y2
z2
x2
+ 2 + 2 = 1.
2
a
b
c
157. There is a famous way to parameterize a sphere called stereographic projection. We will work with
the unit sphere x2 + y 2 + z 2 = 1. Draw a line from a point (a, b) in the xy-plane to the north pole
(0, 0, 1). This line intersects the sphere in a point (x, y, z). This gives a parameterization of the sphere
by points in the plane.
(a) Which point corresponds to the south pole?
(b) Which points correspond to the equator?
(c) Do we get all the points of the sphere by this parameterization?
(d) Which points correspond to the upper hemisphere?
(e) Which points correspond to the lower hemisphere?
(f) (Bonus!) Can you write explicitly the parameterization x = x(a, b), y = y(a, b), z = z(a, b)?
(Hint: it may be useful to try these first for a circle and the real line instead of a sphere and the plane).
32
158. Obtain a parameterization of the surface obtained by rotating the curve x2 z = 1 for x > 0 about the
x-axis.
159. Give a parameterization of the circle of radius a centered at the point (x0 , y0 , z0 ) and in the plane
parallel to two given unit vectors ~u and ~v such that ~u · ~v = 0.
160. A torus (doughnut) is constructed by rotating a small circle of radius a in a large circle of radius b
about the origin. The small circle is in a (rotating) vertical plane through the origin and the large
circle is in the xy-plane. See the figure below. Parameterize the torus by the following method.
z
y
r
θ
s
φ
x
(a) Parameterize the large circle.
(b) For a typical point on the large circle, find two unit vectors which are perpendicular to one another
and in the plane of the small circle at that point. Use these vectors to parameterize the small
circle relative to its center.
(c) Combine your answers to parts (a) and (b) to parameterize the torus.
(d) Use Maple to plot this.
Vector Fields; 17.1
161.
´ Sketch the graphs of the following vector fields.
(a) F~ (x, y) = 2bi + 3bj
(b) F~ (x, y) = ybi
(c) F~ (~r) = 2~r
(d) F~ (x, y) = (x + y)bi + (x − y)bj
162. In Calculus I we saw that given a differentiable function, f , we can obtain a new function f ′ by
differentiation. For functions of many variables we use the gradient as a higher dimensional analogue
to derivative. However, ∇f is not a scalar function, but rather a vector field. For each of the functions
in Problem 19, sketch the gradient field. Hint: if you use their contour diagrams you can do this
without doing any computations whatsoever!
33
The Flow of a Vector Field
163.
164.
´ Sketch the flow for each of the vector fields in Problem 161.
´ For each of the following vector fields, a flow is given. Sketch the field and the flow. Then find the
system of differential equations associated with the field and verify that the flow satisfies the system.
(a) ~v = ybi + xbj;
(b) ~v = ybi − xbj;
(c) ~v = xbi + ybj;
(d) ~v = xbi − ybj;
x(t) = a(et + e−t ),
x(t) = a sin t,
t
y(t) = a(et − e−t ).
y(t) = a cos t.
x(t) = ae ,
y(t) = bet .
x(t) = aet ,
y(t) = be−t .
165. Consider a model for two interacting populations of foxes and rabbits. Let f (t) and r(t) represent the
fox population and the rabbit population as functions of time. Their interaction is govened by the
system
r′ (t) = ar − crf, f ′ (t) = −bf + krf
where a, b, c, and k are positive constants. Find a vector field whose flow is the set of solutions to this
system.
The Idea of a Line Integral; 17.2
166. Consider Rthe vectorRfield F~ shown Rbelow, together with the paths C1 , C2 and C3 . Arrange the line
integrals C1 F~ · d~r, C2 F~ · d~r, and C3 F~ · d~r in ascending order.
00
11
00
11
1111111111111111111111111111
0000000000000000000000000000
0000000000000000000000000000
1111111111111111111111111111
1111111111111111111111111111
0000000000000000000000000000
0000000000000000000000000000
1111111111111111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
C2
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
00
11
000000000000000
0000000000000000000000000000
1111111111111111111111111111
00111111111111111
11
11
00
00
11
001010
11
1010
1010
1010 C1
1010
00
11
001010
11
C3
11
00
00
11
167. ´ For each of the following figures, say whether the line integral of the pictured vector field over the
given curve is positive, negative, or zero.
(a)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
(b)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
(c)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
(d)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
34
168. ´ For each of the given vector fields below, say whether F~ has positive, negative, or zero circulation
around the curve C shown in the figure below. (The two curved segments are circular arcs centered at
the origin). You might find it helpful to first sketch the vector field.
y
2
1
x
1
2
-1
-2
(a) F~ (x, y) = xbi + ybj
(b) F~ (x, y) = −ybi + xbj
(c) F~ (x, y) = ybi − xbj
(d) F~ (x, y) = x2bi
169. ´ Draw anR oriented curve C and a vector field F~ along C that is not always perpendicular to C, but
for which C F~ · d~r = 0.
Computing Line Integrals over Parameterized Curves; 17.2
170. ´ Compute the line integral of the given vector field along the given path.
(a) F~ (x, y) = x2bi + y 2bj and C is the line from the point (1, 2) to the point (3, 4).
(b) F~ (x, y) = ln ybi + ln xbj and C is the curve given parametrically by ~r(t) = 2tbi + t3bj for 2 ≤ t ≤ 4.
(c) F~ (x, y) = exbi + eybj and C is the part of the ellips x2 + 4y 2 = 4 joining the point (0, 1) to the point
(2, 0) in the clockwise direction.
(d) F~ (x, y) = xybi + (x − y)bj and C is the triangle joining the points (1, 0), (0, 1) and (−1, 0) in the
clockwise direction.
b for 1 ≤ t ≤ 2.
b and C is given by ~r(t) = tbi + t2bj + t3 k
(e) F~ (x, y, z) = xbi + 2zybj + xk
171. ´ Consider the vector field F~ = −ybi + xbj. Let C be the unit circle oriented counterclockwise.
(a) Show that F~ has constant magnitude of 1 on the circle C.
(b) Show that F~ is always tangent to the circle C.
R
(c) Show that C F~ · d~r = Length of C.
(d) Explain why this is.
35
Gradient Fields and Conservative Fields; 17.3
172. Decide whether or not the following vector fields could be gradient fields. Justify your answer.
(a) F~ (x, y) = xbi
z
bi + √ y
bj + √ x
b
k
(b) F~ (x, y, z) = − √
2
2
2
2
x +z
x +z
x2 + z 2
~
(c) G(x,
y) = (x2 − y 2 )bi − 2xybj
1
b
(d) F~ (r) = 3 ~r, where ~r = xbi + ybj + z k.
r
173. The statement below is FALSE. Explain why or give a counterexample.
R
“If C F~ · d~r = 0 for one particular closed path C, then F~ is conservative.”
174. In this problem, we see how the Fundamental Theorem for Line Integrals can be derived from the
Fundamental Theorem for ordinary definite integrals. Suppose that (x(t), y(t)), a ≤ t ≤ b is a
parameterization of C which has endpoints P = (x(a), y(a)) and Q = (x(b), y(b)). The values of f
along C are given by the single variable function h(t) = f (x(t), y(t)).
(a) Show that
h′ (t) = fx (x(t), y(t))
dy
dx
+ fy (x(t), y(t)) .
dt
dt
See Problem 152
(b) Use the Fundamental Theorem of Calculus applied to h(t) to show that
R
C
∇f · d~r = f (Q) − f (P ).
2
2
175. Suppose that ∇f = 2xex sin ybi + ex cos ybj. Find the change in f between (0, 0) and (1, π2 ) in two
ways:
(a) By computing a line integral.
(b) By computing f .
176. Suppose a particle subject to a force F~ (x, y) = ybi − xbj moves along the arc of the unit circle, centered
at the origin, that begins at (−1, 0) and ends at (0, 1) (i.e. clockwise).
(a) Find the work done by F~ . Explain the sign of your answer.
(b) Is F~ conservative? Explain.
177. The line integral of F~ = (x + y)bi + xbj along each of the following paths is 3/2.
(i) The path (t, t2 ), with 0 ≤ t ≤ 1.
(ii) The path (t2 , t), with 0 ≤ t ≤ 1.
(iii) The path (t, tn ), with n > 0 and 0 ≤ t ≤ 1.
Verify this in two ways:
(a) Using a parameterization to compute the line integral.
(b) Using the Fundamental Theorem of Calculus for Line Integrals.
36
Nonconservative Fields and Green’s Theorem; 17.4
178. Draw the vector field F~ (x, y) = −ybi + xbj. We have seen this field before and have shown that it
is nonconservative. Here is another way of seeing this. Suppose F~ were the gradient field for some
function f (x, y). Draw and label what the contours of f would have to look like, and explain why it
would not be possible for f to have a single value at any given point.
179. Find f from ∇f :
(a) ∇f = 2xybi + (x2 + 8y 3 )bj
b
(b) ∇f = (yzexyz + z 2 cos(xz 2 ))bi + (xzexyz )bj + (xyexyz + 2xz cos(xz 2 ))k
180. Decide whether the given vector field is the gradient of a function f . If so, find such an f . If not,
explain why not.
(a) F~ (x, y) = ybi + ybj.
(b) F~ (x, y) = (x2 + y 2 )bi + 2xybj.
(c) F~ (x, y) = (2xy 3 + y)bi + (3x2 y 2 + x)bj.
(d) F~ (x, y, z) =
b
i
x
+
b
j
y
b
+ kz .
181. Suppose F~ (x, y) = xbj. Show that the line integral of F~ around a closed curve in the xy-plane measures
the area of the region enclosed by the curve.
182. Calculate the area of the asteroid x2/3 +y 2/3 = a2/3 . Hint: Parameterize by x = a cos3 t, y = a sin3 t, 0 ≤
t ≤ 2π, and use Problem 181, above.
8
6
4
2
–8
–6
–4
–2
0
2
4
6
8
–2
–4
–6
–8
Figure 1: The asteroid x2/3 + y 2/3 = 82/3
37
The Flux Integral; 17.5
183. Arrange the following flux integrals,
and Si are the following surfaces:
Z
Si
b
~ with i = 1, 2, 3, 4, in ascending order if F~ = −bi − bj + k
F~ · dA,
• S1 is a horizontal square of side length 1, oriented upward with one corner at (0, 0, 2) and above
the first quadrant of the xy-plane.
• S2 is a horizontal square of side length 1, oriented upward with one corner at (0, 0, 3) and above
the third quadrant of the xy-plane.
√
• S3 is a square of side length 2 in the xz-plane with one corner at the origin, one edge along the
positive x-axis, one along the negative z-axis,and oriented in the negative y-direction.
√
• S4 is a square of side length 2 with one corner at the origin, one edge along the positive y-axis,
one corner at the point (1, 0, 1), and oriented upwards.
184. Let S be the cube with side 2, faces parallel to the coordinate planes, and centered at the origin.
b out of S by computing the
(a) Calculate the total flux of the constant vector field ~v = −bi + 2bj + k
flux through each face separately.
b
(b) Calculate the flux out of S for any constant vector field ~v = abi + bbj + ck.
(c) Do your answers above make sense?
185. Explain why if F~ has constant magnitude of 1 on S and is everywhere normal to S and in the direction
of orientation, then
Z
S
~ = Area of S.
F~ · dA
186. A fluid is flowing along in a cylindrical pipe of radius a running in the bi direction. The velocity of the
fluid at a distance r from the center of the pipe is ~v = u(1 − r2 /a2 )bi.
(a) What is the significance of the constant u?
(b) What is the velocity of the fluid at the wall of the pipe?
(c) Find the flux through a circular cross-section of the pipe.
Calculating Flux Integrals; 17.5
187. ´ Compute the flux of the given vector field, F~ through the given surface S.
b S is the rectangle z = 4, 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, oriented in the positive
(a) F~ = zbi + ybj + 2xk;
z-direction.
b S is the rectangle y = −1, 0 ≤ x ≤ 2, 0 ≤ z ≤ 4, oriented in the
(b) F~ = x2bi + (x + ey )bj − k;
negative y-direction.
b S is the region in the plane z = x + y above the rectangle 0 ≤ x ≤
(c) F~ = (x − y)bi + zbj + 3xk;
2, 0 ≤ y ≤ 3, oriented upward.
(d) F~ = ~r; S is the surface z = x2 + y 2 , oriented downward, above the disk x2 + y 2 ≤ 1.
b S is the surface y = x2 +z 2 , with x2 +z 2 ≤ 1, oriented in the positive y-direction.
(e) F~ = ybi+bj−xz k;
(f) F~ = xbi + ybj; S is the part of the surface z = 25 − (x2 + y 2 ) above R where R is the disc of radius
5 centered at the origin.
b S is as in part (f).
(g) F~ = cos(x2 + y 2 )k;
p
b S is the cone z = x2 + y 2 for 0 ≤ z ≤ 6.
(h) F~ = −xzbi − yzbj + z 2 k;
38
The Divergence of a Vector Field; 17.6
188. Draw two vector fields whose divergence everywhere is
(a) positive.
(b) negative.
(c) zero.
189. For each of the vector fields below, calculate div F~ and sketch F~ .
~r
(a) F~ (~r) = 3 (in 3-space), ~r 6= ~0.
r
b
(b) F~ (x, y, z) = z k.
190. Find the divergence of the given vector field.
(a) F~ (x, y) = (x2 − y 2 )bi + 2xybj
~r
(b) F~ (~r) = (in 3-space), ~r 6= ~0
r
~
(c) F (x, y) = −xbi + ybj
(d) F~ (x, y) = −ybi + xbj
b
(e) F~ (x, y, z) = (−x + y)bi + (y + z)bj + (−z + x)k
191. For each of the following vector fields, F~ , find the flux of F~ through the box of side length c in the
first octant (positive x, positive y, positive z) with side length c. Then use that answer to find div F~
at the origin.
(a) F~ = xbi
b
(b) F~ = 2bi + ybj + 3k
b
(c) F~ = xbi + ybj + 0k
192. Show that if ~a is a constant vector and f (x, y, z) is a scalar-valued function, and if F~ (x, y, z) = f (x, y, z)~a,
then div F~ = (∇f ) · ~a.
39
The Divergence Theorem; 17.6
193. Compute
R
S
~ in two ways if possible: directly and using the Divergence Theorem.
F~ · dA
(a) F~ (~r) = ~r and S is the cube enclosing the volume 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2.
(b) F~ (x, y, z) = ybj and S is a vertical cylinder of height 2, with its base a circle of radius 1 on the
xy-plane, centered at the origin. S includes the disks at the top and bottom of the cylinder.
b and S is a square pyramid with base on the xy-plane of side length 1 and
(c) F~ (x, y, z) = −zbi + xk
height 3.
R
b
~ where F~ = x2bi + (y − 2xy)bj + 10z k
194. Use the Divergence Theorem to evaluate the flux integral S F~ · dA,
and S is the sphere of radius 5 centered at the origin, oriented outward.
195. Recall that a function φ(x, y, z) is said to be harmonic in a region if div ( grad φ) = 0 at every point
in the region. This equation is also written ∇2 φ = 0, because div ( grad φ) = ∇ · (∇φ). Recall that, as
∂b
∂ b
∂ b
a vector operator, ∇ = ∂x
i + ∂y
j + ∂z
k. Show that
∇2 φ(x, y, z) =
∂ 2 φ ∂ 2 φ ∂ 2φ
+ 2 + 2.
∂x2
∂y
∂z
196. Show that linear functions are harmonic.
R
~ = 0 for
197. Use the Divergence Theorem to show that if φ is harmonic in a region W , then S ∇φ · dA
every closed surface S in W such that the volume enclosed by S lies completely within W .
198. Show that a nonconstant harmonic function φ cannot have a local minimum. (Hint: see your notes).
199. Show that if φ is a harmonic function, then div (φ grad φ) = k grad φk2 .
200. Use the Divergence Theorem to show that if φ and ψ are harmonic functions in a region W , then
Z
Z
~
~
φ∇ψ · dA = − ψ∇φ · dA
S
S
for every closed surface S in W such that the volume enclosed by S lies completely within W .
40
The Curl of a Vector Field; 17.7
201.
Compute the curl of the following vector fields:
(a) F~ (x, y, z) = (x2 − y 2 )bi + 2xybj
~r
(b) F~ (~r) = (in 3-space), ~r 6= ~0
r
b
~
(c) F (x, y, z) = x2bi + y 3bj + z 4 k
2
b
(d) F~ (x, y, z) = exbi + cos ybj + ez k
b
(e) F~ (x, y, z) = 2yzbi + 3xzbj + 7xy k
b
(f) F~ (x, y, z) = (−x + y)bi + (y + z)bj + (−z + x)k
b
(g) F~ (x, y, z) = (x + yz)bi + (y 2 + xyz)bj + (zx3 y 2 + x7 y 6 )k
202. Assume that f (x, y, z) is a scalar function with continuous second
partial derivatives. Use the FundaR
mental Theorem of Calculus for Line Integrals to show that C ∇f · d~r = 0 for any smooth closed path
C. Deduce that curl ∇f = ~0.
203. If F~ is any vector field whose components have continuous second partial derivatives, show that
div curl F~ = 0.
204. For any constant vector field ~c, and any vector field F~ , show that div (F~ × ~c) = ~c · curl F~ .
205. For ~c a constant vector field and F~ any vector field, show that curl (F~ + ~c) = curl F~ .
206. For φ a scalar function and F~ a vector field, show that curl (φF~ ) = φ curl F~ + (∇φ) × F~ .
207. Show that if φ is a harmonic function, then ∇φ is both curl free and divergence free.
Stokes’ Theorem; 17.7
208. Compute the given line integrals using Stokes’ Theorem.
R
b and C is a circle of radius 2 around the y-axis at y = 1 with
(a) C F~ · d~r where F~ = −zbi + ybj + xk
orientation (given by the right hand rule) in the positive y-direction. (i.e., as you look from the
origin, down the y-axis at C, the curve is oriented clockwise.)
R
(b) C F~ · d~r where F~ = r1 ~r and C is the path consisting of straight line segments from (1, 0, 1) to
(1, 0, 0) to (0, 0, 1) back to (1, 0, 1).
R
~ = 0 for any surface which is the complete boundary
209. Use Stokes’ Theorem to show that S curl F~ · dA
surface of a volume V . Deduce that div curl F~ = 0.
~ such that curl G
~ = ybi + xbj? How do you know?
210. Is there a vector field G
211. Determine whether vector potentials for F~ exist, and if so, find one.
b
(a) F~ = 2xbi + (3y − z 2 )bj + (x − 5z)k
b
(b) F~ = x2bi + y 2bj + z 2 k
41
.
42
43
8600
8600
8800
+ 10320
10000
9000
+ 9772
9600
9800
9400
9200
9600
9400
9000
9000
9200
+ 9731
9600
9000
8800
9400
8600
8400
9200
8200
8000
7800
Figure 3: Use this page for Problem 98