Download F(x, y, z)

Notice also that the gradient vectors are long where the level curves are close to each
other and short where the curves are farther apart. That's because the length of the gradient vector is the value of the directional derivative of f and closely spaced level curves
indicate a steep graph.
0
A vector field F is called a conservative vector field if it is the gradient of some scalar
function, that is, if there exists a function f such that F = Vf. In this situation f is called
a potential function for F.
Not all vector fields are conservative, but such fields do arise frequently in physics. For
example, the gravitational field F in Example 4 is conservative because if we define
f(x, y, z)
Vf(x,y,z)
=
af i
ax
mMG
2
YX +y2+Z2
1
+ af j + af k
ay
az
i
-mMGx
(x2
=
+
-mMGy
+ y2 + Z2)3/2
.
+
+ y2 + z2)3/2 J
(x2
k
-mMGz
+ y2 + Z2)3/2
(x2
In Sections 17.3 and 17.5 we will learn how to tell whether or not a given vector field is
conservative.
§
EXERCISES
1-10 Sketch the vector field F by drawing a diagram like
Figure 5 or Figure 9.
,
,
---,
,
...---.---- ... ,
r~---'
•......•......
I. F(x, y)
3. F(x,y)
[II F(x,y)
=
=
+ j)
~(i
+
yi
yi
=
2. F(x,y)
4. F(x, y) = (x - y) i
~j
+
xj
R+J2
7. F(x, y, z)
=
k
8. F(x, y, z)
=
-y k
= i + xj
6. F(x,y)=
+
~
+
y-
=
.....----,
.----, ,, ,
•.......---- , ,
•.......----, ,
..----, , ,
~ ---, ,
t
•..•.•.
.•......
t
-----
9. F(x, y, z) = x k
10. F(x, y, z)
••..
•..
yi-xj
yx-
xj
••.•...
•..•.•.
•.••...
•..•.•.
t
t
t
t
t
---
j - i
......• '-:.,
'"
r
r
r
r
r
r
r
r
r
r
',. ...•...•
"'"
f
f
f
f
f
f
f
f
f
f
\\\
i I
-5 J J
,
11-14 Match the vector fields F with the plots labeled I-IV.
Give reasons for your choices.
[ill
F(x,y)
/
/
'/1'
!
\ •. . , ,
r r
I
,
,
\
\
I
•
•
,
,
,, - - , ,
5
I I , ".- - '" , \ \
,/
•........•.....
---.. ,
•...•.
"''''''''''''' "''''''''''~
'-'-'-'-'- '-'-'-'-'.....•.•.••.....•.....•.....•.....•.....•.....•.....•.....•
,
/////
/////
/' /' /' /' /' /' /' /' /' /'
"""""
""""
"'''''''''''''""'''''''''''''
\
,
,./'/
,!/.r..--- •...•....
,,
. ,
,
---- - •... , --------,/ ,
,
/
.....•..••..•.•.
\ \ , ....
- - ""
?I
/.-.------
...• / ....
, ...• \ ! "" ~
, , \ f , , ,
i r ,
, J
, ,
\ ,, ,
". I \ '" '" '"
-..--- ....
....•...•
-..
......•......•.•..•• ....••. .....•.....•......•......•......•.
•.....
•....•...•...•...
/////
/////
/////
/////'.
= (y,x)
12. F(x, y)
=
(1, sin y)
13. F(x,y)
= (x -
14. F(x,y)
= (y,
2,x
I/x)
+
1)
15-18 Match the vector fields F on
Give reasons for your choices.
15. F(x,y,z)
= i+
2j
+
3k
~3
with the plots labeled I-IV.
16. F(x,y,z)
= i+
2j
+
zk
[j1J
F(x, y, z)
=
18. F(x,y,z)
+yj +3k
xi
+ yj + zk
=xi
", ,, \ \
\
· ...•
"'\1
"
...•..........•.......•.
,/
j I
~JII'1,/,./- ~
1
,
1
/---_-....--",! i, \ \ '
r
, '~IJt""Y"""
---_."
----+-~,/ '\..
..•,.../,
.•
'
,
1
,/
•......•......
-//,/1
\
, ,I
r
,/
'''''///'/'
'''''/.//'
,///'
\
"
/
./
~
I'
/
/'
/
I'
/
,
"
" ,
"
,/ / ,/
,
,/,//,/"
,/,//,/"
,
"
4
,
,
-4
,
4
III
\ \
,"-,\
•.....•...
"
1//'/'
1
\
"
/',...
-4
4
.,.
..-
,/
I I
...•.....•.
\ \
.....•
F(x, y)
=
(y2 - 2xy) i
+
(3xy - 6x2)j
Explain the appearance by finding the set of points (x, y)
such that F(x, y) = O.
Let F(x) = (r2 - 2r)x, where x = (x, y) and r = I x I. Use a
CAS to plot this vector field in various domains until you can
see what is happening. Describe the appearance of the plot
and explain it by finding the points where F(x) = O.
~20.
21. f(x, y)
~
f(x,y,z)
25-26
In(x
=
+
)x2
22. f(x, y) = x
2y)
+
U
y2
+
Z2
24. f(x, y, z)
Find the gradient vector field Vf of
25. f(x, y)
~
=
=
x2 - Y
f
=
e-/3x
x cos(y/z)
and sketch it.
26. f(x, y)
=
)x2
+
y2
27-28 Plot the gradient vector field of f together with a contour
map of f. Explain how they are related to each other.
29-32 Match the functions
vector fields (labeled I-IV).
f
with the plots of their gradient
Give reasons for your choices.
"
\
J
\ \ \
-4
19. If you have a CAS that plots vector fields (the command
is fieldplot in Maple and PlotVectorField in
Mathematica), use it to plot
"
I
,./''/11
~
,
r !I
" ,
""' .. ,
•.. .. ,
•.. .. .. .
--
,...
- •..•....•
- -~ ...•....•
- - ---..•....•
-- -~ ...•
....•
..- •.. ~
•...•...- •...... - - •...•...•...- .,.•...•... - - -
/
- -.,
,
/'
...•
~ ...•
,
,"
"
33. A particle moves in a velocity field Vex, y) = (x2, X + y2).
If it is at position (2, 1) at time t = 3, estimate its location at
time t = 3.01.
34. At time t = I, a particle is located at position (1, 3). If it
moves in a velocity field
~
The flow lines (or streamlines) of a vector field are the paths
followed by a particle whose velocity field is the given vector
field. Thus the vectors in a vector field are tangent to the flow
lines.
(a) Use a sketch of the vector field F(x, y) = xi - y j to
draw some flow lines. From your sketches, can you guess
the equations of the flow lines?
(b) If parametric equations of a flow line are x = x(t),
y = yet), explain why these functions satisfy the differential equationsdx/dt
= x and dy/dt = -yo Then solve
the differential equations to find an equation of the flow
line that passes through the point (1, 1).
36. (a) Sketch the vector field F(x, y) = i + x j and then sketch
some flow lines. What shape do these flow lines appear
to have?
(b) If parametric equations of the flow lines are x = x(t),
y = yet), what differential equations do these functions
satisfy? Deduce that dy/dx = x.
(c) If a particle starts at the origin in the velocity field given
by F, find an equation of the path it follows.
fo F(r(t))
l
fe F . dr
=
• r'(t) dt
Finally, we note the connection between line integrals of vector fields and line integrals
of scalar fields. Suppose the vector field F on 1R3is given in component form by the equation F = Pi + Q j + R k. We use Definition 13 to compute its line integral along C:
L
F . dr
=
=
=
s:
s:
s:
F(r(t))
(P i
. r'(t) dt
+ Q j + R k) . (x'(t) i + y'(t) j + z'(t) k) dt
+ Q(x(t), y(t), z(t)) y'(t) + R(x(t), y(t), z(t)) z'(t)] dt
[p(x(t), y(t), z(tY)x'(t)
Ie F . dr
=
L
P dx
+ Q dy + R dz
+ z dy + x dz in Example 6 could be expressed as
For example, the integral Je y dx
Je F . dr where
§
EXERCISES
1-16 Evaluate the line integral, where C is the given curve.
I. Se y3 ds,
C: x = t3, Y = t, 0 ,,:; t ,,:; 2
2. Jexy
C: x
ds,
rn Je xy4 ds,
5.
Je
-
10.
0,,:; t":; 1
C is the right half of the circle x2
4. Je x sin y ds,
(X2y3
= t2, Y = 2t,
9. Sexy3
C: x
Je xyz2
+
y2 = 16
[IQ
Je xe
dy,
12.
J~(2x
+
13. Sex2y~dz,
6. Je sin x dx,
C is the arc of the curve x
=
y4 from (1, -1) to (1, 1)
14. Sezdx
C: x
C consists of line segments from
(0, 0) to (2, 0) and from (2, 0) to (3, 2)
8.
JexvY
=
3t, 0 ,,:; t ,,:; 7r/2
(x - y) dy,
dx + 2yvlx dy,
Cconsistsoftheshortestarcofthe
circle x2 + y2 = 1 from (1,0) to (0, 1) and the line segment
from (0, 1) to (4, 3)
to (1, 6, 4)
ds,
C is the line segment from (0, 0, 0) to (1, 2, 3)
vIx from (1, 1) to (4, 2)
+
t, z
Yz
=
dx
= 4 cos
ds,
C is the arc of the curve y
[L) Je xy
t, y
C is the line segment from (-1,5,0)
C is the line segment from (0, 3) to (4, 6)
vIx)
ds,
= 4 sin
C:x=t3,
+
=
C: x = t, Y = t2,
9z) ds,
xdy
t2, Y
y=t,
Z
z=t2,
=
t3, 0 ,,:; t ,,:; 1
O":;t":;
1
+
=
ydz,
t3, Z = t2, 0 ,,:; t ,,:; 1
15. Je (x + yz) dx + 2x dy + xyz dz,
C consists of line
segments from (1, 0, 1) to (2, 3, 1) and from (2,3, 1) to
(2,5,2)
16. Je x2 dx
(0,0,0)
+ y2 dy + Z2 dz, C consists of line segments from
to (1, 2, -1) and from (1, 2, -1) to (3, 2, 0)
UII Let
F be the vector field shown in the figure.
(a) If C1 is the vertical line segment from ( - 3, - 3) to (- 3, 3),
determine whether fe, F . dr is positive, negative, or zero.
(b) If C2 is the counterclockwise-oriented
circle with radius 3
and center the origin, determine whether fe, F . dr is positive, negative, or zero.
y
//'~-
//'/
!I ,,-/' 2
I !
I
T
I
,,
3 L2'
\
\
\
'\,
'"
,
.jo
.....•
'-"'-\
"-
'"
"-
\
\
\
i
\
-.JI 0
j
~
" -i-
.•...
•......•.....
25. fe x sin (y
Y = t3, Z
-
-~
-2
.-
2J
I
...- ,/ I
•..•.. //,/
I
\
\
,
27. F(x, y) = (x - y) i + xy j,
C is the arc of the circle x2 + y2
clockwise from (2, 0) to (0, -2)
~x
I
28. F(x, y)
19. F(x, y) = xy i + 3y2 j,
r(t) = llt4i + t3j,
O'S t 'S I
20. F(x,y,z)
= (x + y)i + (y - z)j + z2k,
2
r(t) = t i + t3 j + t2 k, 0 'S t 'S 1
= sin xi + cas y j + xz k,
i - t2 j + t k, 0 'S t'S 1
22. F(x,y,z)
= zi + yj - xk,
r(t) = t i + sin t j + cos t k,
=
t2,
0
'S
t
=
I
+
+
=
4 traversed counter-
h
j,
+y
x from (-1,2) to (1,2)
x
2
30. (a) Evaluate the line integral fe F . dr, where
F(x, y, z) = xi - z j + y k and C is given by
r(t) = 2ti + 3tj - t2k, -1 'S t'S I.
(b) Illustrate part (a) by using a computer to graph C and
the vectors from the vector field corresponding to t = :±: I
and :±:~ (as in Figure 13).
Find the exact value of fex3y2zds,
where C is the curve with
parametric equations x = e-r cos 4t, y = e-r sin 4t, z = e-r,
'S t'S 27T.
o
32. (a) Find the work done by the force field F(x, y) = x2 i + xy j
on a particle that moves once around the circle x2 + y2 = 4
oriented in the counterclockwise direction.
l!m
(b) Use a computer algebra system to graph the force field and
circle on the same screen. Use the graph to explain your
answer to part (a).
~
A thin wire is bent into the shape of a semicircle x2 + y2 = 4,
x "" O. If the linear density is a constant k, find the mass and
center of mass of the wire.
'S 7T
23-26 Use a calculator or CAS to evaluate the line integral correct
to four decimal places.
23. fe F . dr, where F(x, y) = xy i
r(t) = er i + e-r' j, I 'S t 'S 2
t, Y
29. (a) Evaluate the line integral fe F . dr, where
F(x, y) = ex-I i + xy j and C is given by
r(t) = t2 i + t3 j, 0 'S t 'S I.
(b) Illustrate part (a) by using a graphing calculator or computer to graph C and the vectors from the vector field
corresponding to t = 0, 1/ ft, and 1 (as in Figure 13).
l!m31.
19-22 Evaluate the line integral Jc F . dr, where C is given by the
vector function r(t).
i
= ~
x +y
C is the parabola y
~
y, z)
=
Use a graph of the vector field F and the curve C to guess
whether the line integral of F over C is positive, negative, or zero.
Then evaluate the line integral.
~
= t3
= t2,
z) ds, where C has parametric equations x
t , 0 'S t 'S 5
l!m 27-28
18. The figure shows a vector field F and two curves C1 and C2•
Are the line integrals of F over C1 and C2 positive, negative,
or zero? Explain.
r(t)
=
x sin y k
4
26. fe ze-xy ds, where C has parametric equations x
Z = e-r, 0 'S t 'S I
..-.r-,//
'"''''
InJ F(x,
+
+
7T
--.~"-'\.
....•
,-
24. fe F . dr, where F(x, y, z) = y sin z i + z sin xj
and r(t) = cos t i + sin t j + sin 5t k, 0 'S t 'S
+ sin y j
and
34. A thin wire has the shape of the first-quadrant part of the circle
with center the origin and radius a. If the density function is
p(x, y) = kxy, find the mass and center of mass of the wire.
35. (a) Write the formulas similar to Equations 4 for the center of
mass (x, y, z) of a thin wire in the shape of a space curve C
if the wire has density function p(x, y, z).
(b) Find the center of mass of a wire in the shape of the helix
x = 2 sin t, y = 2 cos t, z = 3t, 0 :;;:;t :;;:;27T, if the density
is a constant k.
36. Find the mass and center of mass of a wire in the shape of the
helix x = t, Y = cos t, z = sin t, 0 :;;:;t :;;:;27T, if the density at
any point is equal to the square of the distance from the origin.
37. If a wire with linear density p(x, y) lies along a plane curve C,
its moments of inertia about the x- and y-axes are defined as
38. If a wire with linear density p(x, y, z) lies along a space curve
C, its moments of inertia about the x-, y-, and z-axes are
defined as
=
fe (y2
+
Z2)p(X,
y, z)
Iy
=
fe (x2
+
Z2)p(X,
y, z) ds
Iz = fe (x2
+
y
(meters)
C
ds
I
y, z) ds
0
Find the work done by the force field F(x, y) = xi
in moving an object along an arch of the cycloid
r(t) = (t - sin t) i + (I - cos t) j, 0 :;;:;t :;;:;27T.
40. Find the work done by the force field F(x, y)
on a particle that moves along the parabola y
(-I, I) to (2,4).
=
+
(y
x sin y i
from
+
+
~
/'
.?'
1/
./
./
./
/'"'
x
1
(meters)
2) j
y j
= x2
41. Find the work done by the force field
F(x, y, z) = (y + z, x + z, x + y) on a particle that moves
along the line segment from (1,0,0) to (3,4,2).
42. The force exerted by an electric charge at the origin on a
charged particle at a point (x, y, z) with position vector
r = (x, y, z) is F(r) = Kr/I r 13 where K is a constant. (See
Example 5 in Section 17.1.) Find the work done as the particle
moves along a straight line from (2,0,0) to (2, 1,5).
l1!] A
160-lb man carries a 25-lb can of paint up a helical staircase
that encircles a silo with a radius of 20 ft. If the silo is 90 ft
high and the man makes exactly three complete revolutions,
how much work is done by the man against gravity in climbing
to the top?
44. Suppose there is a hole in the can of paint in Exercise 43 and
9 lb of paint leaks steadily out of the can during the man's
ascent. How much work is done?
45. (a) Show that a constant force field does zero work on a
particle that moves once uniformly around the circle
x2 + y2 = I.
./
11
/
/
7
C
1
~
kx, where k is a
46. The base of a circular fence with radius 10 m is given by
x = 10 cos t, Y = 10 sin t. The height of the fence at position
(x, y) is given by the function hex, y) = 4 + 0.01(x2 - y2), so
the height varies from 3 m to 5 m. Suppose that I L of paint
covers 100 m2. Sketch the fence and determine how much paint
you will need if you paint both sides of the fence.
/
y2)p(X,
=
47. An object moves along the curve C shown in the figure from
(1,2) to (9, 8). The lengths of the vectors in the force field F
are measured in newtons by the scales on the axes. Estimate
the work done by F on the object.
Find the moments of inertia for the wire in Example 3.
Ix
(b) Is this also true for a force field F(x)
constant and x = (x, y)?
48. Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in
the plane perpendicular to the wire and whose center is the axis
of the wire (as in the figure). Ampere's Law relates the electric
current to its magnetic effects and states that
feB . dr
=
/LoI
where I is the net current that passes through any surface
bounded by a closed curve C, and /Lo is a constant called the
permeability of free space. By taking C to be a circle with
radius r, show that the magnitude B = 1 B I of the magnetic
field at a distance r from the center of the wire is
/LoI
B=27Tr
which says that the work done by the force field along C is equal to the change in kinetic
energy at the endpoints of C.
Now let's further assume that F is a conservative force field; that is, we can write
F = Vf. In physics, the potential energy of an object at the point (x, y, z) is defined as
P(x, y, z) = -f(x, y, z), so we have F = - VP. Then by Theorem 2 we have
w=
LF'
which says that if
of a conservative
remains constant.
the vector field is
§
dr = -
L
\7p. dr = -[P(r(b»
- P(r(a»J
= P(A) - P(B)
an object moves from one point A to another point B under the influence
force field, then the sum of its potential energy and its kinetic energy
This is called the Law of Conservation of Energy and it is the reason
called conservative.
EXERCISES
I. The figure shows a curve C and a contour map of a function
whose gradient is continuous. Find fe Vf' dr.
f
6. F(x, y)
=
(3x2 - 2y2) i
[L] F(x, y)
=
(yeX
+
sin y) i
[TIJ The
2. A table of values of a function f with continuous gradient is
given. Find fe Vf' dr, where C has parametric equations
0
1
2
0
1
6
4
1
3
5
7
2
8
2
9
~
3-10 Determine whether or not F is a conservative vector field.
If it is, find a function f such that F = Vf.
3. F(x,y)
=
(2x - 3y)i
4. F(x,y)
=
eXcosyi
+
(-3x
+ eXsinyj
+
4y - 8)j
+
(4xy
+
(eX
+
+
3) j
x cos y) j
figure shows the vector field F(x, y) = (2xy, x2) and
three curves that start at (1,2) and end at (3, 2).
(a) Explain why
F . dr has the same value for all three
curves.
(b) What is this common value?
Ie
12-18 (a) Find a function f such that F = 'Vf and (b) use
part (a) to evaluate Jc F . dr along the given curve C.
12. F(x, y) = xZ i + yZ j,
C is the arc of the parabola y
13. F(x,y)
C: r(t)
=
xyZi
=
(t
+
+
=
26. Let F = 'Vf, where f(x, y) = sin(x - 2y). Find curves C1
and C2 that are not closed and satisfy the equation.
(a)
2xz from (-1,2) to (2, 8)
~
xZyj,
+
sin ~1Tt, t
cos ~1Tt),
0,,;; t,,;; I
f. F·
c.
dr = 0
(b)
f. F·
dr = I
C1
Show that if the vector field F = Pi + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then
Z
-y--z
i + 2y arctan xj,
1+x
C: r(t) = tZ i + 2t j, 0,,;; t ,,;; 1
14. F(x, y)
=
[ill F(x,
y, z) = yz i + xz j + (xy + 2z) k,
C is the line segment from (1,0, -2) to (4, 6, 3)
+
i)i
+ 1, z
16. F(x, y, z) = (2xz
C: x = tZ, Y = t
+
=
28. Use Exercise 27 to show that the line integral
Jc y dx + x dy + xyz dz is not independent of path.
2xy j + (xZ + 3zZ) k,
2t - I, 0,,;; t";; 1
29-32 Determine whether or not the given set is (a) open,
(b) connected, and (c) simply-connected.
17. F(x,y, z) = yZ cos z i + 2xycos zj - xyz sin zk,
C: r(t) = tZ i + sin t j + t k, 0,,;; t ,,;; 7T
18. F(x,y,z)
= eYi + xeYj + (z + l)e'k,
C: r(t) = t i + tZ j + t3 k, 0,,;; t ,,;; 1
19-20 Show that the line integral is independent of path and
evaluate the integral.
19. Jc 2x sin y dx
+
(xZ cos y - 3i)
Jc (2yZ
-
12x3y3)
+
dx
dy,
+
2y3/Z i
(4xy - 9x4yZ) dy,
22. F(x,y)
=
e-Yi - xe-Yj;
3x.jy
j;
P(I,I),
y
L J I
~ ! J
\ \
\
.•...•
--...
\
"
-.
\
."
----
-t>-/,/
11m 25.
I
I ,r
, ..
,
-
,
/'
I
.. ,
I
..- .-.-. •......
y
r i
,
i i
-----... -. .. ,
-t>_/,/
I
/'
I
-
,
!
I
,\ \\
, \x
.•...•
"
."
\
\
\
I
\
!
r i
\ \ I
~ ! J I ,r ..L J I I /.....---
"'
\
"
•.....
24.
, i \i
F(x, y) =
-y i
Z
+
xj
z·
cr
= ~
Q(2,4)
P(O, I), Q(2,0)
/.....---
+ yZ ,,;; I or 4 ,,;; xZ + yZ ,,;; 9}
F(r)
23-24 Is the vector field shown in the figure conservative?
Explain.
[ill
I xZ
(b) Show that Jc F . dr is not independent of path.
[Hint: Compute Jc, F . dr and Jc, F . dr, where CI
and Cz are the upper and lower halves of the circle
xZ + yZ = 1 from (1,0) to (-1,0).]
Does this contradict
Theorem 6?
21-22 Find the work done by the force field F in moving an
object from P to Q.
=
32. {(x, y)
x + y
(a) Show that ap/ay = aQ/ax.
C is any path from (1, I) to (3, 2)
21. F(x, y)
II < x2 + y2 < 4}
IllJ Let
C is any path from (-1,0) to (5, 1)
20.
31. {(x, y)
, ..
\
"' "
•.....x
.-. •......
.--
If F(x, y) = sin y i + (1 + x cos y) j, use a plot to guess
whether F is conservative. Then determine whether your
guess is correct.
for some constant c, where r = xi + y j + z k. Find the
work done by F in moving an object from a point PI
along a path to a point Pz in terms of the distances dl and
dz from these points to the origin.
(b) An example of an inverse square field is the gravitational field F = -(mMG)r/1
r 13 discussed in Example 4
in Section 17.1. Use part (a) to find the work done by
the gravitational field when the earth moves from aphelion (at a maximum distance of 1.52 X 108 km from
the sun) to perihelion (at a minimum distance of
1.47 X 108 km). (Use the values m = 5.97 X IOz4 kg,
M = 1.99 X 1030 kg, and G = 6.67 X 10-11 N·mz/kgz.)
(c) Another example of an inverse square field is the electric
force field F = eqQr/1 r 13 discussed in Example 5 in
Section 17.1. Suppose that an electron with a charge of
-1.6 X 10-19 C is located at the origin. A positive unit
charge is positioned a distance 10-12 m from the electron
and moves to a position half that distance from the electron. Use part (a) to find the work done by the electric
force field. (Use the value e = 8.985 X 109.)
We now easily compute this last integral using the parametrization
r(t) = a cos t i + a sin t j, 0 :;;;;t :;;;;
27T. Thus
F.
ie
dr
=
iF'
dr
=
C'
[27T
Jo
F(r(t)) . r'(t) dt
(-a sin t)(-a sin t) + (acos t)(acos t)
-------------dt
o
a2 cos2t + a2 sin2t
i
27T
=
given by
i271"
=
dt =
27T
0
We end this section by using Green's Theorem to discuss a result that was stated in the
preceding section.
SKETCH OF PROOF OF THEOREM
an open simply-connected
derivatives, and that
17.3.6 We're assuming that F = Pi + Q j is a vector field on
region D, that P and Q have continuous first-order partial
ap
aQ
ay
ax
If C is any simple closed path in D and R is the region that C encloses, then Green's
Theorem gives
fe F . dr
=
fe P dx + Q dy
=
If (~~ - ~:)
dA
=
R
Sf
0 dA
=
0
R
A curve that is not simple crosses itself at one or more points and can be broken up into
a number of simple curves. We have shown that the line integrals of F around these
simple curves are all 0 and, adding these integrals, we see that
F . dr = 0 for any
closed curve C. Therefore
F . dr is independent of path in D by Theorem 17.3.3. It
follows that F is a conservative vector field.
0
Ie
Ie
§
EXERCISES
1-4 Evaluate the line integral by two methods: (a) directly and
(b) using Green's Theorem.
I.
1e (x
- y) dx + (x + y) dy,
with center the origin and radius 2
e is the circle
,h
2
2. jexy dx + x dy,
e is the rectangle with vertices (0, 0), (3, 0), (3, I), and (0, 1)
rn ~exy
e
4.
dx + X2y3 dy,
is the triangle with vertices (0, 0), (I, 0), and (1, 2)
1e x dx
+ y dy,
e consists
of the line segments from (0, 1)
to (0,0) and from (0,0) to (1,0) and the parabola y = 1 - x2
from (I, 0) to (0, 1)
6. Ie x2 y2 dx + 4xy3 dy,
e is the triangle with vertices (0, 0), (1, 3), and (0, 3)
m j~(y
+
eJx)dx + (2x + cosy2)dy,
boundary of the region enclosed by the parabolas
x2 and x = y2
e is the
y
8.
=
Ie xe-2x
dx + (x' + 2X2y2) dy,
e is the boundary of the region between the circles
x2 + y2 = J and x2 + y2 = 4
[!J Iey3
dx - x3 dy,
10. Ie sin y dx
e is the circle x2 + y2 = 4
+ x cos y dy, e is the ellipse x2 + xy + y2
=
1
Ie
5-10 Use Green's Theorem to evaluate the line integral along the
given positively oriented curve.
5. Ie e dx + 2xe dy,
e is the square with sides x = 0, x = 1, Y = 0, and y = 1
Y
Y
11-14 Use Green's Theorem to evaluate
F . dr. (Check the
orientation of the curve before applying the theorem.)
JY),
II. F(x,y) = (J; + y3,x2 +
e consists of the arc of the curve y = sin x from (0, 0) to (7T, 0)
and the line segment from (7T, 0) to (0, 0)
12. F(x, y) = (y2 cos X, x2 + 2y sin x),
C is the triangle from (0, 0) to (2, 6) to (2, 0) to (0, 0)
(eX + x2y, eY - xyZ),
C is the circle x2 + y2 = 25 oriented clockwise
13. F(x,y)
=
In(x2
(y - 3)2
+
14. F(x, y) = (y -
(x - 2)2
[ill]
+
=
y2), 2 tan-1 (y/x),
C is the circle
I oriented counterclockwise
Let D be a region bounded by a simple closed path C in the
xy-plane. Use Green's Theorem to prove that the coordinates
of the centroid (:t,)i) of Dare
15-16 Verify Green's Theorem by using a computer algebra system to evaluate both the line integral and the double integral.
Q(x, y) = x2eY,
C consists of the line segment from (-I, I) to (I, I) followed
by the arc of the parabola y = 2 - x2 from (I, I) to (-I, 1)
15. P(x, y) = y2e",
3
16. P(x, y) = 2x - X y5,
C is the ellipse 4x2 +
3
yZ
Q(x, y) = X y8,
= 4
23. Use Exercise 22 to find the centroid of a quarter-circular
region of radius a.
24. Use Exercise 22 to find the centroid of the triangle with
vertices (0,0), (a, 0), and (a, b), where a > 0 and b > O.
[!L] Use
25. A plane lamina with constant density p(x, y) = p occupies a
region in the xy-plane bounded by a simple closed path C.
Show that its moments of inertia about the axes are
18. A particle starts at the point (-2,0), moves along the x-axis
26. Use Exercise 25 to find the moment of inertia of a circular
Green's Theorem to find the work done by the force
F(x, y) = x(x + y) i + xy2 j in moving a particle from the
origin along the x-axis to (1, 0), then along the line segment
to (0, I), and then back to the origin along the y-axis.
to (2, 0), and then along the semicircle y = J4="XT to the
starting point. Use Green's Theorem to find the work done on
this particle by the force field F(x, y) = (x, x3 + 3xyZ).
19. Use one of the formulas in (5) to find the area under one arch
of the cycloid x
~
(c) Find the area of the pentagon with vertices (0,0), (2, I),
(1, 3), (0, 2), and ( -I, I).
=
t - sin t, y
=
I - cos t.
20. If a circle C with radius 1 rolls along the outside of the
circle x2 + yZ = 16, a fixed point P on C traces out a
curve called an epicycloid,
with parametric equations
x = 5 cos t - cos 5t, Y = 5 sin t - sin St. Graph the epicycloid and use (5) to find the area it encloses.
[Ih] (a)
If C is the line segment connecting the point (Xl, YI) to the
point (X2, Yz), show that
fcxdY
- ydx
= XlYZ - XZYI
(b) If the vertices of a polygon, in counterclockwise order,
are (Xl, YI), (X2, Y2), ... , (x"' Yn), show that the area of
the polygon is
A
=
H(XlY2
+
- X2YI)
+
(X2Y3 - X3Yz)
(Xn-1Y" - XnYn-l)
=================~
+
+ ...
(XnYI - XlYn)]
CURL AND
disk of radius a with constant density p about a diameter.
(Compare with Example 4 in Section 16.5.)
Se
InJ If F is the
vector field of Example 5, show that
F . dr = 0
for every simple closed path that does not pass through or
enclose the origin.
28. Complete the proof of the special case of Green's Theorem
by proving Equation 3.
29. Use Green's Theorem to prove the change of variables
formula for a double integral (Formula 16.9.9) for the case
where f(x, y) = 1:
If
dx dy
=
Jf
I :~~: ~~ I du dv
Here R is the region in the xy-plane that corresponds to the
region S in the uv-plane under the transformation given by
x = g(u, v), y = h(u, v).
[Hint: Note that the left side is A(R) and apply the first part
of Equation 5. Convert the line integral over aR to a line integral over as and apply Green's Theorem in the uv-plane.]
DIVERGENCE
In this section we define two operations that can be performed on vector fields and that
playa basic role in the applications of vector calculus to fluid flow and electricity and magnetism. Each operation resembles differentiation, but one produces a vector field whereas
the other produces a scalar field.
§
EXERCISES
1-8 Find (a) the curl and (b) the divergence of the vector field.
rn F(x,
y, z)
=
xyz i - x2y k
13-18 Determine whether or not the vector field is conservative.
If it is conservative, find a function f such that F = Vf.
13. F(x, y, z)
=
y2z3 i
+
2xyz3 j
+ 3xy2z2
14. F(x, y, z)
=
xyz2 i
+
X2yz2
j
+
+
(x2
2yz) j
[ill F(x,
4. F(x, y, z) = xeY j
5. F(x, y, z)
=
/
yx2
+
+
ye: k
1
y2
(x i +
+ Z2
+ ytan-'(x/z)k
6. F(x,y,z)
=
eXYsinzj
7. F(x, y, z)
=
(In x, In(xy),
8. F(x, y, z)
=
(eX, eXY, eXY')
yj
+
z k)
In(xyz»
t
t
t
=
2xy i
16. F(x, y, z)
=
e' i + j + xe: k
17. F(x,y,z)
=ye-Xi
18. F(x,y,z)
= ycosxyi
U!l Is there
curl G
9-11 The vector field F is shown in the xy-plane and looks the
same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.)
(a) Is div F positive, negative, or zero? Explain.
(b) Determine whether curl F = O. If not, in which direction does
curl F point?
t
+
+
i i i i
2zk
xcosxyj
- sinzk
a vector field G on IR 3 such that
(x sin y, cos y, z - xy)? Explain.
=
20. Is there a vector field G on IR 3 such that
curl G = (xyz, _y2z, yz2)? Explain.
[TIJ
Show that any vector field of the form
23-29 Prove the identity, assuming that the appropriate partial
derivatives exist and are continuous. If f is a scalar field and F, G
are vector fields, then fF, F . G, and F x G are defined by
(fF)(x,
23. div(F
+
+
24. curl(F
25. div(fF)
12. Let f be a scalar field and F a vector field. State whether
each expression is meaningful. If not, explain why. If so, state
whether it is a scalar field or a vector field.
(a) curlf
(b) gradf
(c) div F
(d) curl(gradf)
27. div(F
28. div(Vf
X
y, z) F(x, y, z)
F(x, y, z) . G(x, y, z)
(F X G)(x, y, z)
=
F(x, y, z) X G(x, y, z)
G)
G)
=
f
G)
curl F
V g)
29. curl(curl F)
=
=
F· Vf
+
(Vf)
X
F
0
grad(div F) - V2F
+
(f) grad(div F)
30-32 Let r
(h) grad(div f)
(i) curl(curl F)
(j) div(div F)
30. Verify each identity.
(a) V . r = 3
(c) V2r3 = 12r
xi
curl G
G . curl F - F . curl G
(g) di v(grad f)
=
+
div G
+
curl F
div F
=
+
div F
=
(e) grad F
(I) div(curl(gradf»
= f(x.
=
=f
X
y, z)
(F . G)(x, y, z)
=
26. curl(fF)
(div F)
y2 k
////
t t t t
x
+
e-'j
+
k
11//
t t t t
(k) (gradf)
+
y, z)
X2y2z
k
yj
+
z k and r
=
I r I.
[Q Verify
each identity.
(a) Vr
=
(c) V(l/r)
r/r
=
(b) V
-r/r3
X r =
0
(d) Vln r = r/r2
32. If F = r/ rP, find div F. Is there a value of p for which
div F = o?
33. Use Green's Theorem in the form of Equation 13 to prove
Green's first identity:
ff
fV2gdA
=
fJ(Vg)'
nds
D
-
Sf Vf·
VgdA
D
where D and C satisfy the hypotheses of Green's Theorem
and the appropriate partial derivatives of f and 9 exist and are
continuous. (The quantity V9 . n = Dn 9 occurs in the line integral. This is the directional derivative in the direction of the
normal vector n and is called the normal derivative of g.)
34. Use Green's first identity (Exercise 33) to prove Green's
second identity:
Sf
(fV2g
- gV2f)
dA
=
fc (fV 9 -
gV f) .
n ds
38. Maxwell's equations relating the electric field E and magnetic
field H as they vary with time in a region containing no charge
and no current can be stated as follows:
D
where D and C satisfy the hypotheses of Green's Theorem
and the appropriate partial derivatives of f and 9 exist and are
continuous.
35. Recall from Section 15.3 that a function 9 is called harmonic
on D if it satisfies Laplace's equation, that is, V2g = 0 on D.
Use Green's first identity (with the same hypotheses as in Exercise 33) to show that if 9 is harmonic on D, then ~c Dng ds = O.
Here Dng is the normal derivative of 9 defined in Exercise 33.
36. Use Green's first identity to show that if f is harmonic
on D, and iff(x,y)
= 0 on the boundary curve C, then
il"D I Vf 12 dA = O. (Assume the same hypotheses as in
Exercise 33.)
37. This exercise demonstrates a connection between the curl
vector and rotations. Let B be a rigid body rotating about the
z-axis. The rotation can be described by the vector w = wk,
where w is the angular speed of B, that is, the tangential speed
of any point P in B divided by the distance d from the axis of
rotation. Let r = (x, y, z) be the position vector of P.
(a) By considering the angle (J in the figure, show that the
velocity field of B is given by v = w X r.
(b) Show that v = -wyi
+ wxj.
(c) Show that curl v = 2w.
curIE
I
aH
c
at
= ---
curlH
I aE
=--
c
at
where c is the speed of light. Use these equations to prove the
following:
(a) V
X
(b) V
X
(c) V2E
(V
X
(V
X
I
= ---
c
2
I
E)
= --- 2
H)
= ---
a2E
at2
c
a2E
at2
1
a2H
2
at2
c
[Hint: Use Exercise 29.]
39. We have seen that all vector fields of the form F = Vg
satisfy the equation curl F = 0 and that all vector fields of the
form F = curl G satisfy the equation div F = 0 (assuming
continuity of the appropriate partial derivatives). This suggests
the question: Are there any equations that all functions of the
form f = div G must satisfy? Show that the answer to this
question is "No" by proving that every continuous function f
on ~ 3 is the divergence of some vector field. [Hint: Let
G(x, y, z) = (g(x, y, z), 0, 0), where g(x, y, z) = j~'
f(t, y, z) dt.]
Converting to polar coordinates, we obtain
Io I: )1 + 4r2 r dr d{} = fa27Td{} I: r)l
A =
+ 4r2 dr
27r
The question remains whether our definition of surface area (6) is consistent with the
surface area formula from single-variable calculus (9.2.4).
We consider the surface S obtained by rotating the curve y = j(x), a ~ x ~ b, about
the x-axis, where j(x) ?' 0 and l' is continuous. From Equations 3 we know that parametric equations of S are
+ 1'(x) cos {}j + 1'(x) sin {}k
rx
=
i
re
=
-j(x)
+ j(x) cos {}k
sin {}j
j
r,
re
X
o
=
Irx
X
rei
=
k
1 1'(x) cos {} 1'(x) sin {}
=
-j(x)
j(x)1'(x)
sin {} j(x) cos {}
i - j(x) cos {}j - j(x) sin {}k
+ [j(X)]2COS2{} + [j(x)]2sin2{}
)[j(x)J2[1'(X)]2
= )[j(x)J2[1 + [1'(X)]2] = j(x))l
A
=
II Ir
x
X
re IdA
=
f
+ [1'(x)]2
+ [1'(X)]2 dx d{}
fa27T j(x))l
D
= 27T'
+ [1'(X)]2 dx
fj(x))l
This is precisely the formula that was used to define the area of a surface of revolution in
single-variable calculus (9.2.4).
§
EXERCISES
1-2 Determine whether the points P and Q lie on the given surface.
I. r(u, v) = (2u + 3v, I + 5u - v,2
P(7, 10,4), Q(5, 22, 5)
2. r(u, v) = (u + v, u2 - v,
P(3, -1,5), Q(-I,3,4)
U
+
+
u
+
v)
v2)
~
5. r(x,
e)
=
6. r(x,
e)
= (x, x cos
(x, cos
7.r(u,v)=(u2+1,vJ+l,u+v),
II] r(u,
8. r(u,v)
=
(u
+
v) i
+
(3 - v) j
+
(1
+
4u
+
5v) k
sin
e)
e, x
sin
e)
7-12 Use a computer to graph the parametric surface. Get a printout
and indicate on it which grid curves have u constant and which have
v constant.
3-6 Identify the surface with the given vector equation.
v)
e,
= (u
+
2 2
V,U ,V ),
-I~u~I,
-l~v~l
-1 ~ u ~ 1, -1 ~ v ~ 1
9. r(u, v) = (u cos v, u sin v, u5),
-1 ~ u ~ 1, 0 ~ v ~
27T'
10. r(u, v) = (cos u sin v, sin u sin v, cos v
o "S U "S 2 7T, 0.1 "S v "S 6.2
II. x
o
sin v, y = cos u sin 4v, z
"S U "S 27T, -7T/2 "S v "S 7T/2
=
=
+
In tan (v/2) ),
19-26 Find a parametric representation
~
sin 2u sin 4v,
for the surface.
The plane that passes through the point (I, 2, - 3) and
contains the vectors i + j - k and i - j + k
20. The lower half of the ellipsoid 2x2
21. The part of the hyperboloid
+
4/
+
Z2
Z2
=
1 that lies to the
+/ -
2
x
=
1
right of the xz-plane
22. The part of the elliptic paraboloid x +
in front of the plane x = 0
13-18 Match the equations with the graphs labeled I-VI and
give reasons for your answers. Determine which families of grid
curves have u constant and which have v constant.
14. r(u, v)
=
ucosvi
15. r(u, v)
=
sin v i
+ usinvj
+
+
cos u sin 2v j
sinuk,
+
sin
-7T"S
U
U
1m The part
of the sphere x2 +
cone z = x2 + y2
+ y2 +
=2
sin 2v k
+
25. The part of the cylinder y2
16. x
=
(I - u)(3
Y
=
(l - u)(3
z = 3u
+
+
+
cos v) cos
47TU,
cos v) sin
47TU,
planes x
~
(l - u) sin v
+ 2z2
=
4 that lies
Z2
=
4 that lies above the
Z2
=
16 that lies between
J
24. The part of the sphere x2
the planes z = -2 and z
"S 7T
+
y2
y2
=
0 and x
=
The part of the plane z
x2 + y2 = I
Z2
=
16 that lies between the
5
=
x
+
3 that lies inside the cylinder
[ill] 27-28
Use a computer algebra system to produce a graph that
looks like the given one.
~
29. Find parametric equations for the surface obtained by rotating
the curve y = e -", 0 "S x "S 3, about the x-axis and use them
to graph the surface.
~
30. Find parametric equations for the surface obtained by rotating
the curve x = 4y2 - y4, -2 "S Y "S 2, about the y-axis and
use them to graph the surface.
~
31. (a) What happens to the spiral tube in Example 2 (see Fig-
ure 5) if we replace co< u by sin u and sin u by cos u?
(b) What happens if we replace cos u by cos 2u and sin u
by sin 2u?
~
32. The surface with parametric equations
x
=
2cos fJ + rcos(fJ/2)
y
=
2 sin fJ + rcos(fJ/2)
z = r sin(fJ/2)
where - ~ "S r "S ~ and 0 "S fJ "S 2 7T,is called a Mobius
strip. Graph this surface with several viewpoints. What is
unusual about it?
33-36 Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that
graphs parametric surfaces, use a computer to graph the surface
and the tangent plane.
y=3u2,
ITI;]x=u+v,
35. r(u, v) = U 2 i
+
z=u-v;
+
2u sin v j
51. (a) Use the Midpoint Rule for double integrals (see Section 16.1) with six squares to estimate the area of the
surface z = I/(l + x2 + y2), 0 "" X "" 6,0"" y "" 4.
[ill]
(b) Use a computer algebra system to approximate the
surface area in part (a) to four decimal places. Compare
with the answer to part (a).
(2,3,0)
u = I, v = 0
u cos v k;
36. r(u, v) = uv i + u sin v j + v cos uk;
u = 0, v =
part of the plane 3x
first octant
+
38. The part of the plane 2x
+
cylinder x2
+
39. The surfacez
y2
=
=
+
2y
z
=
Find the exact area of the surface z
1 "" x "" 4, 0 "" y "" 1.
+
5y
z
=
+
y3/2),
0"" X "" 1,0""
y "" 1
~
[ill]
part of the surface z = xy that lies within the cylinder
y2 = 1
+
~
3x + 2y2 that lies above the
triangle with vertices (0, 0), (0, 1), and (2, 1)
42. The part of the surface z = 1
43. The part of the hyperbolic
between the cylinders x2
44. The part of the paraboloid
cylinder y2
+
Z2
+
paraboloid z = y2 - x2 that lies
+ y2 = 1 and x2 + y2 = 4
x = y2
+
Z2
~
that lies inside the
= 9
45. The part of the surface y = 4x
+
Z2
~
46. The helicoid (or spiral ramp) with vector equation
I1LJ The
z =
+
u sin v j
+
v k, 0 "" u "" 1, 0 "" v ""
7T
surface with parametric equations x = u2, Y = uv,
0 "" U "" 1, 0 "" v "" 2
~V2,
48-49 Find the area of the surface correct to four decimal places
by expressing the area in terms of a single integral and using your
calculator to estimate the integral.
=
cos(x
2
+
y2) that lies inside the
49. The part of the surface z = e-x'-y' that lies above the disk
x2
[ill] 50.
+
2x
+
3y
+
4y2,
(a) Show that the parametric equations x = a sin u cos v,
y = b sin u sin v, z = c cos u, 0 "" u "" 7T, 0 "" V "" 27T,
represent an ellipsoid.
(b) Use the parametric equations in part (a) to graph the ellipsoid for the case a = 1, b = 2, c = 3.
(c) Set up, but do not evaluate, a double integral for the surface area of the ellipsoid in part (b).
equations x = a cosh u cos v,
y = b cosh u sin v, z = c sinh u, represent a hyperboloid
of one sheet.
(b) Use the parametric equations in part (a) to graph the
hyperboloid for the case a = I, b = 2, c = 3.
(c) Set up, but do not evaluate, a double integral for the surface area of the part of the hyperboloid in part (b) that lies
between the planes z = - 3 and z = 3.
IEJ Find
48. The part of the surface z
cylinder x2 + y2 = 1
+
56. (a) Show that the parametric
that lies between the
planes x = 0, x = 1, z = 0, and z = I
r(u, v) = u cos v i
I
of the surface with parametric equations x = au cos v,
y = bu sin v, z = u2, 0 "" U "" 2,0"" v "" 27T.
(b) Eliminate the parameters to show that the surface is an
elliptic paraboloid and set up another double integral for
the surface area.
(c) Use the parametric equations in part (a) with a = 2 and
b = 3 to graph the surface.
(d) For the case a = 2, b = 3, use a computer algebra
system to find the surface area correct to four decimal
places.
10 that lies inside the
9
~(X3/2
=
54. (a) Set up, but do not evaluate, a double integral for the area
r(u, v) = (1 + v, u - 2v, 3 - 5u + v) that is given by
0"" u "" 1,0 "" v "" 1
x2
[ill] 53.
6 that lies in the
40. The part of the plane with vector equation
I1IJ The
Find the area of the surface with vector equation
r(u, v) = (cos3u cos3v, sin3u cosJv, sinJv), 0"" u "" 7T,
o "" V "" 27T. State your answer correct to four decimal
places.
7T
37-47 Find the area of the surface.
Ill;] The
[ill] 52.
y2 "" 4
Find, to four decimal places, the area of the part of the
surface z = (l + x2)/(l
+ y2) that lies above the square
I x I + I y I "" 1. Illustrate by graphing this part of the surface.
the area of the part of the sphere x2
lies inside the paraboloid z = x2 + y2.
+
y2
+
Z2
=
4z that
58. The figure shows the surface created when the cylinder
y2 + Z2 = 1 intersects the cylinder x2 +
area of this surface.
Z2
= 1. Find the
59. Find the area of the part of the sphere x2
lies inside the cylinder x2 + y2 = ax.
+
y2
+
Z2
=
a2 that
60. (a) Find a parametric representation for the torus obtained
by rotating about the z-axis the circle in the xz-plane with
center (b, 0, 0) and radius a < b. [Hint: Take as parameters the angles () and a shown in the figure.]
ffi
(b) Use the parametric equations found in part (a) to graph
the torus for several values of a and b.
(c) Use the parametric representation
surface area of the torus.
================::::::EJ
from part (a) to find the
SU RFAC E INT EG RALS
The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length. Suppose f is a function of three variables
whose domain includes a surface S. We will define the surface integral off over S in such
a way that, in the case where f(x, y, z) = 1, the value of the surface integral is equal to the
surface area of S. We start with parametric surfaces and then deal with the special case
where S is the graph of a function of two variables.
v
We first assume that the parameter domain D is a rectangle and we divide it into subrectangles Rij with dimensions flu and flv. Then the surface S is divided into corresponding
patches Sij as in Figure 1. We evaluate f at a point Pi! in each patch, multiply by the area
flSij of the patch, and form the Riemann sum
Au
~
0
u
L L f(P/J)
i~l
[r
flSij
j~l
Then we take the limit as the number of patches increases and define the surface integral
of f over the surface S as
Sf f(x,
y,
z) dS
=
,}~!!l",
i~l
j~
f(P/J)
flSij
s
Notice the analogy with the definition of a line integral (17.2.2) and also the analogy with
the definition of a double integral (16.1.5).
To evaluate the surface integral in Equation 1 we approximate the patch area flSij by the
area of an approximating parallelogram in the tangent plane. In our discussion of surface
area in Section 17.6 we made the approximation
§
EXERCISES
I. Let S be the boundary surface of the box enclosed by the
planes x = 0, x = 2, y = 0, y = 4, z = 0, and z = 6. Approximate
e-O.I(x+y+:) dS by using a Riemann sum as in Definition I, taking the patches Sij to be the rectangles that are the
faces of the box S and the points p;[ to be the centers of the
rectangles.
13.
.11
2. A surface S consists of the cylinder x2 + y2 = I, -I ~z ~ I,
together with its top and bottom disks. Suppose you know that
f is a continuous function with
Estimate the value of .III f(x, y, z) dS by using a Riemann sum,
taking the patches S;j to be four quarter-cylinders and the top
and bottom disks.
14 .
f(x,y,
z) = g(Jx2 + yl + Z2), where 9 is a
function of one variable such that g(2) = - 5. Evaluate
f(x, y, z) dS, where S is the sphere x2 + yl + Z2 = 4.
+
= x2
Z2
that lies inside the
.Ill dS,
S is the part of the sphere x2 + l + Z2 = 4 that lies
inside the cylinder Xl + l = I and above the xy-plane
[ill .U~(xlz + y2Z)
dS,
S is the hemisphere
+
Xl
l + Z2
=
4, z
?
0
J:1~xz
16.
dS,
S is the boundary of the region enclosed by the cylinder
l + Zl = 9 and the planes x = 0 and x + y = 5
17 .
.I}, (z +
3. Let H be the hemisphere x2
+ y2 + Zl = 50, z ? 0, and
suppose f is a continuous function with f(3, 4, 5) = 7,
f(3, -4,5) = 8, f( -3,4,5) = 9, and f( -3, -4,5) = 12.
By dividing H into four patches, estimate the value of
JJH f(x, y, z) dS.
.I}, y dS,
S is the part of the paraboloid y
cylinder x2 + Z2 = 4
18 .
xly) dS,
S is the part of the cylinder l + Z2 = I that lies between the
planes x = 0 and x = 3 in the first octant
.IJ~(X2+ y2 + z2)dS,
S is the part of the cylinder x2 + l = 9 between the planes
z = 0 and z = 2, together with its top and bottom disks
[!] Suppose that
19-30 Evaluate the surface integral .IJs F . dS for the given vector
field F and the oriented surface S. In other words, find the flux of F
across S. For closed surfaces, use the positive (outward) orientation.
.11
5-18 Evaluate the surface integral.
rn .I1x2yz
[I!]
dS,
S is the part of the plane z
rectangle [0, 3] X [0,2]
6.
7.
8.
10.
II.
12.
I
+
2x
+
3y that lies above the
JJsxy
dS,
S is the part of the plane x
first octant
i - xzeY j + z k,
S is the part of the plane x + y + z = I in the first octant and
has downward orientation
+
y
+
z = I that lies in the
+ y j + Z4 k,
S is the part of the cone z = ~
with downward orientation
22. F(x, y, z) = xi
JJs y dS,
=
hX3/2 + y3/2),
0 ~
X~
I, 0 ~ y ~ 1
.11
yzdS,
S is the surface with parametric equations x
z = u cos v, 0 ~ u ~ 1, 0 ~ v ~ 7T/2
23. F(x, y, z) = xi
=
ul, Y
=
u sin v,
x2 + y2 dS,
S is the helicoid with vector equation
r(u, v) = u cos vi + u sin v j + v k, 0 ~ u ~ I, 0 ~ v ~
JJs
o~
21. F(x, y, z) = xzeY
JJ~yz
.IJ~,II
F(x, y, z) = xy i + yz j + zx k, S is the part of the
paraboloid z = 4 - Xl - y2 that lies above the square
x ~ I, 0 ~ y ~ I, and has upward orientation
20. F(x, y, z) = y i + x j + Z2 k,
S is the helicoid of Exercise 10 with upward orientation
dS,
S is the triangular region with vertices (I, 0, 0), (0, 2, 0),
and (0, 0, 2)
S is the surface z
9.
=
JJsz dS,
S is the surface x
=
y
+
Z2
= Xl
2z2,
+
y2
that lies between the
0 ~ Y ~ I, 0 ~ z ~ 1
y j
+
+
y2
7T
Zl
=
9
F(x, y, z) = y j - z k,
S consists of the paraboloid y = Xl
and the disk x2 + Z2 ~ 1, y = 1
y ~ 0 with upward
+
Z2,
0 ~ Y ~ 1,
26. F(x, y, z) = xy i + 4xl j + yz k, S is the surface z = xeY,
x ~ I, 0 ~ Y ~ 1, with upward orientation
o~
=
z k,
+
orientation
~
Xl Z2
+
Xl
24. F(x,y,z)
= -yi
+ xj + 3zk,
S is the hemisphere z = J~I-6---x-2---y-2,
+
dS,
S is the part of the cone
planes z = I and z = 3
S is the sphere
beneath the plane z
I
27. F(x, y, z) = xi + 2y j + 3z k,
S is the cube with vertices (± I, ± I, ± I)
28. F(x, y, z) = xi + y j + 5 k, S is the boundary of the region
enclosed by the cylinder x2 + Z2 = 1 and the planes y = 0
and x + y = 2
39. (a) Give an integral expression for the moment of inertia lz
about the z-axis of a thin sheet in the shape of a surface S
if the density function is p.
(b) Find the moment of inertia about the z-axis of the funnel
in Exercise 38.
40. Let S be the part of the sphere x2 + y2 + Z2 = 25 that lies
above the plane z = 4. If S has constant density k, find
(a) the center of mass and (b) the moment of inertia about
the z-axis.
29. F(x, y, z) = x2 i + y2 j + Z2 k, S is the boundary of the
solid half-cylinder 0 :;;;z :;;; J1="Y2, 0 :;;; x :;;; 2
30. F(x, y, z) = y i + (z - y) j + x k,
S is the surface of the tetrahedron with vertices (0, 0, 0),
(1,0,0), (0,1,0), and (0,0, I)
41. A fluid has density 870 kg/m3
and flows with velocity
v = z i + l j + x2 k, where x, y, and z are measured in
meters and the components of v in meters per second. Find
the rate of flow outward through the cylinder x2 + y2 = 4,
0:;;; z :;;; I.
Ji~
lim 31.
Evaluate
xyz dS correct to four decimal places, where S is
the surface z = xy, 0 :;;;x :;;; 1,0 :;;;y :;;; 1.
lim 32.
Find the exact value of
Exercise 31.
limn.
Find the value of f.1~x2y2z2dS correct to four decimal places,
where S is the part of the paraboloid z = 3 - 2x2 - y2 that
lies above the x y-plane.
lim 34.
Find the flux of
.11 x
2
yz dS, where S is the surface in
F(x, y, z) = sin(xyz) i
+
x2y j
+
43. Use Gauss's Law to find the charge contained in the solid
hemisphere x2 + y2 + Z2 :;;; a2, z ;;;;:0, if the electric field is
Z2ex/5 k
across the part of the cylinder 4y2 + Z2 = 4 that lies above
the xy-plane and between the planes x = -2 and x = 2 with
upward orientation. Illustrate by using a computer algebra
system to draw the cylinder and the vector field on the same
screen.
35. Find a formula for .IJs F . dS similar to Formula 10 for the
case where S is given by y = h(x, z) and n is the unit normal
that points toward the left.
.11
36. Find a formula for
F . dS similar to Formula 10 for the
case where S is given by x = k(y, z) and n is the unit normal
that points forward (that is, toward the viewer when the axes
are drawn in the usual way).
IE;] Find
the center of mass of the hemisphere x2
z ;;;;:0, if it has constant density.
+
y2
+
Z2 =
a2,
38. Find the mass of a thin funnel in the shape of a cone
z = x2 + y2, 1 :;;;z :;;;4, if its density function is
p(x, y, z) = 10 - z.
.J
================~
42. Seawater has density 1025 kg/m3 and flows in a velocity field
v = y i + x j, where x, y, and z are measured in meters and
the components of v in meters per second. Find the rate of
flow outward through the hemisphere x2 + y2 + Z2 = 9,
z;;;;: O.
44. Use Gauss's Law to find the charge enclosed by the cube
with vertices (± I, ± I, ± I) if the electric field is
~
The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u(x, y, z) = 2y2 + 2z2. Find the rate of
heat flow inward across the cylindrical surface y2 + Z2 = 6,
0:;;; x:;;; 4.
46. The temperature at a point in a ball with conductivity K is
inversely proportional to the distance from the center of the
ball. Find the rate of heat flow across a sphere S of radius a
with center at the center of the ball.
47. Let F be an inverse square field, that is, F(r) = cr /1 r 13 for
some constant c, where r = xi + y j + z k. Show that the
flux of F across a sphere S with center the origin is independent of the radius of S.
STO KES' TH EO REM
Stokes' Theorem can be regarded as a higher-dimensional version of Green's Theorem.
Whereas Green's Theorem relates a double integral over a plane region D to a line integral
around its plane boundary curve, Stokes' Theorem relates a surface integral over a surface
S to a line integral around the boundary curve of S (which is a space curve). Figure 1 shows
§
EXERCISES
rn A hemisphere
10. F(x, y, z) = xy i + 2z j + 3y k, C is the curve of intersection
of the plane x + z = 5 and the cylinder x2 + y2 = 9
H and a portion P of a paraboloid are shown.
Suppose F is a vector field on IRJ whose components have continuous partial derivatives. Explain why
ff
curl F . dS
H
=
ff
curl F . dS
P
and C is the curve of intersection of the plane
x + y + z = 1 and the cylinder x2 + y2 = 9 oriented
counterclockwise as viewed from above.
(b) Graph both the plane and the cylinder with domains
chosen so that you can see the curve C and the surface
that you used in part (a).
(c) Find parametric equations for C and use them to graph C.
Jc
2-6 Use Stokes' Theorem to evaluate
2. F(x,y,z)
= 2ycoszi
S is the hemisphere x2
upward
JJs curl F
+ eXsinzj + xeYk,
+ y2 + Z2 = 9, z ?
3. F(x, y, z) = X2Z2 i + iz2 j + xyz k,
S is the part of the paraboloid z = x2 +
cylinder x2 + y2 = 4, oriented upward
l
ffi
. dS.
0, oriented
ffi
that lies inside the
4. F(x, y, z) = x2 yJz i + sin(xyz) j + xyz k,
S is the part of the cone y2 = x2 + Z2 that lies between the
planes y = and y = 3, oriented in the direction of the
positive y-axis
°
m F(x, y, z)
xyz i + xy j + x2yz k,
S consists of the top and the four sides (but not the bottom)
of the cube with vertices (:t: I, :t: I, :t: I), oriented outward
[Hint: Use Equation 3.]
=
12. (a) Use Stokes' Theorem to evaluate
F . dr, where
F(x, y, z) = x2y i + ~xJ j + xy k and C is the curve of
intersection of the hyperbolic paraboloid z = y2 - x2 and
the cylinder x2 + y2 = 1 oriented counterclockwise as
viewed from above.
(b) Graph both the hyperbolic paraboloid and the cylinder with
domains chosen so that you can see the curve C and the
surface that you used in part (a).
(c) Find parametric equations for C and use them to graph C.
13-15 Verify that Stokes' Theorem is true for the given vector
field F and surface S.
13. F(x,y,z)
= y2i
+ xj + z2k,
S is the part of the paraboloid z
plane z = I, oriented upward
(x + y2)i + (y + Z2)j + (z + x2)k,
C is the triangle with vertices (1,0, 0), (0, 1,0), and (0, 0, 1)
=
9. F(x, y, z) = yz i
C is the circle x2
+ 2xz j + eXY k,
+ y2 = 16, z =
y
+
2z
=
=
2 that lies in the first
that lies below the
2
Z2
=
I, Y
?
0, oriented in the
16. Let C be a simple closed smooth curve that lies in the plane
+
y
+
z
=
I. Show that the line integral
depends only on the area of the region enclosed by C and not
on the shape of C or its location in the plane.
17. A particle moves along line segments from the origin to
the points (1,0,0), (1,2,1), (0,2,1), and back to the
origin under the influence of the force field
F(x, y, z)
5
+z
z) = xi + yj + xyz k,
S is the part of the plane 2x + y
octant, oriented upward
y, z) = y i + z j + x k,
S is the hemisphere x2 + y2 +
direction of the positive y-axis
7-10 Use Stokes' Theorem to evaluate j~ F . dr. In each case C is
oriented counterclockwise as viewed from above.
+
y2
14. F(x,y,
x
8. F(x, y, z) = e-x i + eX j + e' k,
C is the boundary of the part of the plane 2x
in the first octant
+
[ill F(x,
6. F(x, y, z) = eXY cos z i + x2z j + xy k,
S is the hemisphere x = .jl - y2 - Z2, oriented in the direction of the positive x-axis
[Hint: Use Equation 3.]
m F(x,y,z)
= x2
=
Z2 i
+
2xy j
+
4y2 k
Jc (y
+ sin x) dx +
(Z2
+ cosy) dy +
where C is the curve r(t) = (sin t, cos t, sin 2t), 0
[Hint: Observe that C lies on the surface z = 2xy.]
[!!J
20. Suppose Sand C satisfy the hypotheses of Stokes' Theorem
and f, 9 have continuous second-order partial derivatives. Use
Exercises 24 and 26 in Section 16.5 to show the following.
x3dz
"S
t
If S is a sphere and F satisfies the hypotheses of Stokes'
Theorem, show that
curl F . dS = O.
.11
"S
27T.
(a) Jc (f V g) • dr
(b)
.I~(f
(c)
.I~(fVg
Vf)
=
• dr
+
gVf)
=
f.1~(V f X
V g) . dS
0
• dr
=
0
WRITING
PROJECT
• Thephotographshowsa stained-glass
windowat CambridgeUniversityin honorof
GeorgeGreen.
Courtesy of the Masters and Fellows of Gonville and
Caius College. University of Cambridge. England
Although two of the most important theorems in vector calculus are named after George Green
and George Stokes, a third man, William Thomson (also known as Lord Kelvin), played a large
role in the formulation, dissemination, and application of both of these results. All three men
were interested in how the two theorems could help to explain and predict physical phenomena
in electricity and magnetism and fluid flow. The basic facts of the story are given in the margin
notes on pages 1092 and 1129.
Write a report on the historical origins of Green's Theorem and Stokes' Theorem. Explain the
similarities and relationship between the theorems. Discuss the roles that Green, Thomson, and
Stokes played in discovering these theorems and making them widely known. Show how both
theorems arose from the investigation of electricity and magnetism and were later used to study a
variety of physical problems.
The dictionary edited by Gillispie [2] is a good source for both biographical and scientific
information. The book by Hutchinson [5] gives an account of Stokes' life and the book by
Thompson [8] is a biography of Lord Kelvin. The articles by Grattan-Guinness [3] and Gray [4]
and the book by Cannell [I] give background on the extraordinary life and works of Green.
Additional historical and mathematical information is found in the books by Katz [6] and
Kline [7].
I. D. M. Cannell, George Green, Mathematician and Physicist 1793-1841: The Background to
His Life and Work (Philadelphia: Society for Industrial and Applied Mathematics, 2001).
2. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner's, 1974). See the
article on Green by P. J. Wallis in Volume XV and the articles on Thomson by Jed Buchwald
and on Stokes by E. M. Parkinson in Volume XIII.
www.stewartcalculus.com
The Internet is another source of information for this project. Click on History
of Mathematics. Follow the links to the
St. Andrew's site and that of the British
Society for the History of Mathematics.
3. I. Grattan-Guinness, "Why did George Green write his essay of 1828 on electricity and
magnetism?" Amer. Math. Monthly, Vol. 102 (1995), pp. 387-396.
4. J. Gray, "There was a jolly miller." The New Scientist, Vol. 139 (1993), pp. 24-27.
5. G. E. Hutchinson, The Enchanted Voyage and Other Studies (Westport, CT: Greenwood
Press, 1978).
6. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins,
pp. 678-680.
1993),
7. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford
University Press, 1972), pp. 683-685.
8. Sylvanus P. Thompson, The Life of Lord Kelvin (New York: Chelsea, 1976).
the origin. [This is a special case of Gauss's Law (Equation 17.7.11) for a single charge.
The relationship between c and co is c = 1/(47Tco).]
Another application of the Divergence Theorem occurs in fluid flow. Let v(x, y, z) be
the velocity field of a fluid with constant density p. Then F = pv is the rate of flow per
unit area. If Po(xo, Yo, zo) is a point in the fluid and Ba is a ball with center Po and very small
radius a, then div F(P) = div F(Po) for all points in Ba since div F is continuous. We
approximate the flux over the boundary sphere Sa as follows:
ff
/1
1
(
/1/
//1
/
---".,.
/
• P?
/
/'
..-A~~
/ / r
r
1//
/11
r
/1/
§
= x2 i + /
=
fff
div F dV
div F(Po)
1
=
=
fff
div F(Po) dV
lim _(1
a~O
/
V Ba
)
if
=
div F(Po)V(Ba)
F . dS
Sa
Equation 8 says that div F(Po) is the net rate of outward flux per unit volume at Po. (This
is the reason for the name divergence.) If div F(P) > 0, the net flow is outward near P and
P is called a source. If div F(P) < 0, the net flow is inward near P and P is called a sink.
For the vector field in Figure 4, it appears that the vectors that end near PI are shorter
than the vectors that start near PI. Thus the net flow is outward near PI, so div F(PI) > 0
and PI is a source. Near P2, on the other hand, the incoming arrows are longer than the
outgoing arrows. Here the net flow is inward, so div F(P2) < 0 and P2 is a sink. We
can use the formula for F to confirm this impression. Since F = x2 i + y2 j, we have
div F = 2x + 2y, which is positive when y > -x. So the points above the line y = -x
are sources and those below are sinks.
/
FIGURE 4
The vector field F
F . dS
j
EXERCISES
1-4 Verify that the Divergence Theorem is true for the vector field
F on the region E.
OJ F(x,y,z)
= 3xi + xyj
+ 2xzk,
E is the cube bounded by the planes x
y = I, z = 0, and z = 1
=
0, x
2. F(x, y, z) = x2 i + xy j + z k,
E is the solid bounded by the paraboloid z
and the xy-plane
=
=
1, y
4 - x2
=
0,
-
y2
3. F(x, y, z) = xy i + yz j + zx k,
E is the solid cylinder x2 + y2 ,,;; I, 0,,;; z";; 1
4. F(x, y, z) = xi + y j + z k,
E is the unit ball x2 + y2 + Z2
,,;;
y, z) = 3xy2 i + xe' j + Z3 k,
S is the surface of the solid bounded by the cylinder
y2 + Z2 = 1and the planes x = -1 and x = 2
[L] F(x,
8. F(x, y, z) = x3y i - X2y2 j - x2yz k,
S is the surface of the solid bounded by the hyperboloid
x2 + y2 - Z2 = 1 and the planes z = -2 and z = 2
[!J F(x,
y, z) = xy sin z i
S is the ellipsoid x2/a2
10. F(x, y, z) = x2y i
+ xy2 j + 2xyz k,
S is the surface of the tetrahedron bounded by the planes
x = 0, y = 0, z = 0, and x + 2y + z = 2
I
xy2) i + xe-' j + (sin y + x2z) k,
S is the surface of the solid bounded by the paraboloid
z = x2 + y2 and the plane z = 4
II. F(x, y, z) = (cos z
5-15 Use the Divergence Theorem to calculate the surface integral
. dS; that is, calculate the flux of F across S.
.11 F
5. F(x, y, z) = eX sin y i + eX cos y j + yz2 k,
S is the surface of the box bounded by the planes x
x = I, y = 0, y = 1, z = 0, and z = 2
+ cos(xz) j + y cos z k,
+ y2/b2 + Z2/C2 = I
+
~~.4i - X3z2 j + 4xy2z k,
S is the surface of the solid bounded by the cylinder
x2 + y2 = I and the planes z = x + 2 and z =
12. F(x, y, z) =
=
6. F(x,y,z)
= X2z3i + 2xyz3j
+ xz4k,
S is the surface of the box with vertices (± 1, ±2, ±3)
0,
°
4x3zi + 4y3zj + 3z4k,
S is the sphere with radius R and center the origin
13. F(x,y,z)
=
14. F = r/I r I, where r = xi + y j + _z_k_,
5 consists of the hemisphere z = JI - x2
disk x2 + i "S 1 in the xy-plane
[ill] 15.
[ill] 16.
_
-
y2
and the
F(x,y,z)
= e)'tanzi
+ y~
j + xsinyk,
5 is the surface of the solid that lies above the xy-plane
and below the surface z = 2 - x4 - y4, -1 "S x "S I,
-1 "Sy"S
1
Use a computer algebra
F(x, y, z) = sin x cos2y
in the cube cut from the
Y = 7T/2, and z = 7T/2.
surface of the cube.
system to plot the vector field
i + sin3y cos4zj + sin5z cos6x k
first octant by the planes x = 7T/2,
Then compute the flux across the
23. Verify that div E
24. Use the Divergence Theorem to evaluate ffs (2x
where 5 is the sphere x2 + y2 + Z2 = 1.
~ ff
a . n d5
ztan-\y2)i
+ z3ln(x2 + I)j + zk.
Find the flux of F across the part of the paraboloid
x2 + y2 + Z = 2 that lies above the plane z = I and is
oriented upward.
[!!] A
vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F
is positive or negative at PI and at P2•
2
'\.\ \ \ \ I \ \ \\;
, ,
\
-... .......
Pl' ,
- - ---
--- -'-.\
\
I
\
\
--
,
\
,....•
. , , , , -.
I
I
--
, ,....•::
,
I
---
~~~
\
\ \ \ j \ \ \)
(a) Are the points PI and P2 sources or sinks for the vector
field F shown in the figure? Give an explanation based
solely on the picture.
(b) Given that F(x, y) = (x, y2), use the definition of divergence to verify your answer to part (a).
'\ \ \
,t
2
1 1 fir
, , ,, "
, , \pl 1 r I I I /
••..••"
,
1\
- ~ --- -- -" , f;:
,,
\
\\\,
[ill] 21-22
I
-- ~- ---- - -"
,,
/
, ,
1
,
/
I I I /
I 1 1 fI!
Plot the vector field and guess where div F > 0 and
where div F < O. Then calculate div F to check your guess.
21. F(x, y)
=
26. V(E)
(xy, x
+
y2)
2y
+
Z2) d5
=
0, where a is a constant vector
ff
=~
F . dS, where F(x, y, z) = xi
+
yj
+
zk
s
27.
Sf
curl F . dS
=
0
s
28.
=
-. , ,
....• ,
x.
s
jJs F • dS, where
F(x,y.z)
= z2xi + (h3 + tanz)j
+ (x2z + y2)k
and 5 is the top half of the sphere x2 + y2 + Z2 = I.
[Hint: Note that S is not a closed surface. First compute
integrals over 51 and 52, where 51 is the disk x2 + y2 "S I,
oriented downward, and 52 = 5 U 5i-l
....•
+
1:~3
25-30 Prove each identity, assuming that 5 and E satisfy the conditions of the Divergence Theorem and the scalar functions and
components of the vector fields have continuous second-order
partial derivatives.
17. Use the Divergence Theorem to evaluate
18. LetF(x,y,z)
0 for the electric field E(x) =
=
ff
ff
ff
Dnfd5
s
29.
=
fff
2
V fdV
E
(jVg)
. n d5
=
s
30.
fff
(jV2g
+
Vf· Vg) dV
E
(jVg
- gV f) . n d5
fff
=
s
(jV2g
- gV2f)
dV
E
31. Suppose 5 and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that
Sf fnd5
=
S
fff
VfdV
E
These surface and triple integrals of vector functions are
vectors defined by integrating each component function.
[Hint: Start by applying the Divergence Theorem to F = fe,
where e is an arbitrary constant vector.]
32. A solid occupies a region E with surface 5 and is immersed in
a liquid with constant density p. We set up a coordinate
system so that the xy-plane coincides with the surface of the
liquid and positive values of z are measured downward into
the liquid. Then the pressure at depth z is p = pgz, where 9 is
the acceleration due to gravity (see Section 6.5). The total
buoyant force on the solid due to the pressure distribution is
given by the surface integral
F
= -
ff
pn d5
s
where n is the outer unit normal. Use the result of Exercise 31
to show that F = - Wk, where W is the weight of the liquid
displaced by the solid. (Note that F is directed upward
because z is directed downward.) The result is Archimedes'
principle: The buoyant force on an object equals the weight of
the displaced liquid.
=================0
REVI EW
I. What is a vector field? Give three examples that have physical
meaning.
2. (a) What is a conservative vector field?
(b) What is a potential function?
3. (a) Write the definition of the line integral of a scalar function
f along a smooth curve C with respect to arc length.
(b) How do you evaluate such a line integral?
(c) Write expressions for the mass and center of mass of a thin
wire shaped like a curve C if the wire has linear density
function p(x, y).
(d) Write the definitions of the line integrals along C of a
scalar function f with respect to x, y, and z.
(e) How do you evaluate these line integrals?
4. (a) Define the line integral of a vector field F along a smooth
curve C given by a vector function r(t).
(b) If F is a force field, what does this line integral represent?
(c) If F = (P, Q, R), what is the connection between the line
integral of F and the line integrals of the component functions P, Q, and R?
5. State the Fundamental Theorem for Line Integrals.
6. (a) What does it mean to say that Jc F . dr is independent
of path?
(b) If you know that Jc F . dr is independent of path, what can
you say about F?
(c) If F is a velocity field in fluid flow, what are the physical
interpretations of curl F and div F?
10. If F = Pi + Q j, how do you test to determine whether F is
conservative? What if F is a vector field on 1R3?
II. (a) What is a parametric surface? What are its grid curves?
(b) Write an expression for the area of a parametric surface.
(c) What is the area of a surface given by an equation
z = g(x, y)?
12. (a) Write the definition of the surface integral of a scalar function f over a surface 5.
(b) How do you evaluate such an integral if 5 is a parametric
surface given by a vector function r(u, v)?
(c) What if 5 is given by an equation z = g(x, y)?
(d) If a thin sheet has the shape of a surface 5, and the density
at (x, y, z) is p(x, y, z), write expressions for the mass and
center of mass of the sheet.
13. (a) What is an oriented surface? Give an example of a nonorientable surface.
(b) Define the surface integral (or flux) of a vector field F over
an oriented surface 5 with unit normal vector n.
(c) How do you evaluate such an integral if 5 is a parametric
surface given by a vector function r(u, v)?
.
(d) What if 5 is given by an equation z
= g(x, y)?
7. State Green's Theorem.
8. Write expressions for the area enclosed by a curve C in terms
of line integrals around C.
9. Suppose F is a vector field on 1R3.
(a) Define curl F.
(b) Define div F.
Determine
whether
the statement
is true or false. If it is true, explain why.
If it is false, explain why or give an example that disproves the statement.
16. In what ways are the Fundamental Theorem for Line Integrals,
Green's Theorem, Stokes' Theorem, and the Divergence
Theorem similar?
5. If F = Pi + Q j and Pv
conservative.
=
Qx in an open region D, then F is
I. If F is a vector field, then div F is a vector field.
2. If F is a vector field, then curl F is a vector field.
3. If f has continuous partial derivatives of all orders on IR3, then
div(curl \1f) = O.
4. If f has continuous partial derivatives on IR3 and C is any
circle, then
\1f' dr = O.
J~
7. If 5 is a sphere and F is a constant vector field, then
.(I~F·
dS = O.
I. A vector field F, a curve C, and a point P are shown.
(a) Is .I~
F . dr positive, negative, or zero? Explain.
(b) Is div F(P) positive, negative, or zero? Explain.
13-14 Show that F is conservative and use this fact to evaluate
along the given curve.
L F . dr
13. F(x, y) = (4X3y2 - 2xy3) i + (2x4y - 3X2y2 + 4y3)j,
C: r(t) = (t + sin 7Tt) i + (2t + COS 7Tt)j, 0 .:; t':; I
14. F(x,y,z)
= eYi + (xeY + e')j
+ ye'k,
C is the line segment from (0,2,0) to (4,0,3)
15. Verify that Green's Theorem is true for the line integral
Jc xy2 dx - x2 Y dy, where C consists of the parabola y = Xl
from (-1, I) to (J, 1)and the line segment from (J, 1) to
(-1, I).
16. Use Green's Theorem to evaluate Jc JI+X1 dx + 2xydy,
where C is the triangle with vertices (0,0), (1,0), and (1, 3).
2-9 Evaluate the line integral.
2. Jcx ds,
C is the arc of the parabola y
3.
= Xl
J~yz cos x ds,
C: x = t, Y = 3 cos t, z = 3 sin t,
from (0, 0) to (I, 1)
°.:;
t .:;
7T
4. Jc y dx + (x + y2) dy,
C is the ellipse 4x2
counterclockwise orientation
+
9y2 = 36 with
5. .I~y3 dx + x2 dy,
C is the arc of the parabola x
from (0, -I) to (0, 1)
6.
Jc /iY
dx + eY dy + xz dz,
C is given by r(t) = t4 i + t2 j
+
17. Use Green's Theorem to evaluate Jcxlydx
- xyldy,
where C is the circle Xl + yl = 4 with counterclockwise
orientation.
=
I - y2
19. Show that there is no vector field G such that
curl G = 2x i + 3yz j - xz2 k.
20. Show that, under conditions to be stated on the vector fields F
and G,
°.:;
t3 k,
t .:; 1
y2dy
7. Jcxydx
+
+ yzdz,
C is the line segment from (1,0, -1), to (3, 4, 2)
8. Jc F . dr, where F(x, y) = xy i + x2 j and C is given by
r(t) = sin ti + (I + t)j, 0.:; t':; 7T
9. JcF' dr, where F(x,y, z) = e'i + xzj
C is given by r(t) = t2 i + t] j - t k,
°.:;+
(x + y) k and
t':; I
21. If C is any piecewise-smooth simple closed plane curve
and f and 9 are differentiable functions, show that
Jcf(x) dx + g(y) dy = O.
23. If f is a harmonic function, that is, V'2f = 0, show that the line
integral j, dx - Ix dy is independent of path in any simple
region D.
J
10. Find the work done by the force field
F(x, y, z) = z i + x j + Y k in moving a particle from the
point (3, 0, 0) to the point (0, 7T/2, 3) along
(a) a straight line
(b) the helix x = 3 cos t, Y = t, z = 3 sin t
11-12
tion
f
Show that F is a conservative vector field. Then find a funcsuch that F = V' f.
II. F(x, y) = (I
12. F(x,y,z)
=
+
xy)e-"
sinyi
+
i
+
xcosyj
(e'
+
x2e")
- sinzk
j
(b) Find .I~2xe2Ydx
+
(2x2e2Y
+
2ycotz)dy
- y2csc2Zdz.
25. Find the area of the part of the surface z = x + 2y that lies
above the triangle with vertices (0,0), (1,0), and (I, 2).
2
26. (a) Find an equation of the tangent plane at the point (4, -2, I)
to the parametric surface S given by
(b) Use a computer to graph the surface 5 and the tangent
plane found in part (a).
(c) Set up, but do not evaluate, an integral for the surface area
of S.
(d) If
Z2.
x2.
y2
F(x, y, z) = ---?
I + ---?
J + ---?
k
I + xI + yI + zfind
.11 F
Evaluate .I~
F . dr, where C is the curve with initial point
(0, 0, 2) and terminal point (0, 3, 0) shown in the figure.
. dS correct to four decimal places.
27-30 Evaluate the surface integral.
27.
f.l~z dS,
28.
J:l~(x2z + y2Z)
where 5 is the part of the paraboloid z
that lies under the plane z = 4
z
29 .
30.
=
=
x2
+
y2
38. Let
dS, where 5 is the part of the plane
4 + x + y that lies inside the cylinder x2 + y2
=
4
.11 F . dS,
where F(x, y, z) = xz i - 2y j + 3x k and 5 is
the sphere x2 + l + Z2 = 4 with outward orientation
_ (2x3 + 2xy2
F (x,y ) -
- 2y) i + (2y3 + 2x2y
2
X
+
+ 2x) j
2
Y
Evaluate ~e F . dr, where C is shown in the figure.
.11 F
. dS, where F(x, y, z) = x2 i + xy j + z k and 5 is the
part of the paraboloid z = x2 + y2 below the plane z = I
with upward orientation
31. Verify that Stokes' Theorem is true for the vector field
F(x, y, z) = x2 i + y2 j + Z2 k, where 5 is the part of the
paraboloid z = I - x2 - y2 that lies above the xy-plane and
5 has upward orientation.
.11
32. Use Stokes' Theorem to evaluate
curl F . dS, where
F(x, y, z) = x2 yz i + yz2 j + z3exy k, 5 is the part of the
sphere x2 + y2 + Z2 = 5 that lies above the plane z = I, and
5 is oriented upward.
39. Find.ll F . n dS, where F(x, y, z) = xi + y j + z k and 5 is
the outwardly oriented surface shown in the figure (the
boundary surface of a cube with a unit corner cube removed).
Ie
33. Use Stokes' Theorem to evaluate
F . dr, where
F(x, y, z) = xy i + yz j + zx k, and C is the triangle with
vertices (I, 0, 0), (0, I, 0), and (0, 0, I), oriented counterclockwise as viewed from above.
34. Use the Divergence Theorem to calculate the surface integral
F . dS, where F(x, y, z) = x3 i + y3 j + Z3 k and 5 is the
surface of the solid bounded by the cylinder x2 + y2 = I and
the planes z = and z = 2.
.11
a
35. Verify that the Divergence Theorem is true for the vector
field F(x, y, z) = xi + y j + z k, where E is the unit ball
x2 + y2 + Z2 'S I.
F(x, y, z)
=
through the ellipsoid 4x2
xi+yj+zk
(x2 + y2 + z2)3/2
+
9y2
+
6z2
=
36.
37. Let
F(x, y, z)
=
(3x2yz
- 3y) i + (x3z - 3x) j + (x3y + 2z) k
40. If the components of F have continuous second partial derivatives and S is the boundary surface of a simple solid region,
show that
curl F . dS = O.
.11
41. If a is a constant vector, r = xi + y j + z k, and 5 is an oriented, smooth surface with a simple, closed, smooth,
positively oriented boundary curve C, show that
if
s
2a . dS
=
L
(a x r) . dr
================================================1
PRO In EMS
I. Let S be a smooth parametric surface and let P be a point such
at P intersects S at most once. The solid angle D(S) subtended
starting at P and passing through S. Let Sea) be the intersection
the sphere with center P and radius a. Then the measure of the
defined to be
1
D(S)
ID(s)1
I
that each line that starts
by S at P is the set of lines
of D(S) with the surface of
solid angle (in steradians) is
= area of Sea)
a2
=
if
r' n
~
dS
5
where r is the radius vector from P to any point on S, r = 1 r I, and the unit normal vector n is
directed away from P.
This shows that the definition of the measure of a solid angle is independent of the radius a
of the sphere. Thus the measure of the solid angle is equal to the area subtended on a unit
sphere. (Note the analogy with the definition of radian measure.) The total solid angle subtended by a sphere at its center is thus 47T steradians.
2. Find the positively oriented simple closed curve C for which the value of the line integral
Jc (y3 - y) dx - 2x3 dy is a maximum.
3. Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal
vector n =
b, e) and has positive orientation with respect to n. Show that the plane area
enclosed by C is .I~(bz - ey) dx + (ex - az) dy + (ay - bx) dz.
<a,
~
i
4. Investigate the shape of the surface with parametric equations x = sin u, y = sin v,
z = sin(u + v). Start by graphing the surface from several points of view. Explain the
appearance of the graphs by determining the traces in the horizontal planes z = 0, z = ::'::1,
andz=::'::~.
5. Prove the following identity:
V(F . G) = (F . V)G
+
(G . V)F
+
F
X
curl G
+
G
x
curl F
6. The figure depicts the sequence of events in each cylinder of a four-cylinder internal combustion engine. Each piston moves up and down and is connected by a pivoted arm to a rotating
crankshaft. Let pet) and Vet) be the pressure and volume within a cylinder at time t, where
a "S t "S b gives the time required for a complete cycle. The graph shows how P and V vary
through one cycle of a four-stroke engine. During the intake stroke (from CD to ®) a mixture
of air and gasoline at atmospheric pressure is drawn into a cylinder through the intake valve
as the piston moves downward. Then the piston rapidly compresses the mix with the valves
closed in the compression stroke (from ® to ®) during which the pressure rises and the volume decreases. At ® the sparkplug ignites the fuel, raising the temperature and pressure at
almost constant volume to @. Then, with valves closed, the rapid expansion forces the piston
downward during the power stroke (from @ to ®). The exhaust valve opens, temperature and
pressure drop, and mechanical energy stored in a rotating flywheel pushes the piston upward,
forcing the waste products out of the exhaust valve in the exhaust stroke. The exhaust valve
closes and the intake valve opens. We're now back at CD and the cycle starts again.
(a) Show that the work done on the piston during one cycle of a four-stroke engine is
W = Jc P dV, where C is the curve in the PV-plane shown in the figure.
[Hint: Let x(t) be the distance from the piston to the top of the cylinder and note that
the force on the piston is F = AP(t) i, where A is the area of the top of the piston. Then
W = Jc, F . dr, where C1 is given by r(t) = x(t) i, a "S t "S b. An alternative approach is
to work directly with Riemann sums.]
(b) Use Formula 17.4.5 to show that the work is the difference of the areas enclosed by the
two loops of C.