171
PROBLEMS
39
PROBLEMS
origin
Laws ofVector
3-1: Basic
Section
Find an expression for the unit vector directed
the line described
from an arbitrary
point on
toward the
by x
=
z=-3.
Algebra
and
directed toward the
Find an expression for the unit vector
axis at a heighth above the x-y plane
point P located on the z
z
(x, y, -5) in the plane
from an arbitrary
point
3.10
Vector A starts at point (1, -1,-3) and ends at
point
Find a unit vector in the direction
*3.1
1.0).
Given
=
of A.
vectors
=*2 -y3 +2, B =$2-ý+23,
A
4+ý2-22, show
that
C is perpendicular
=
to
both
A
Find a unit vector
3.11
and
=
the area
a4 Given A
B, and
(a) Find
B,if A
Find a relation
b)
to
.5
C
=$B,+2 +2B:
and B
is parallel
to
expressions:
if
A
is
Line
A
2-24,
=*+y2-23,
B
=$2 -Ý4,
and
find the following:
The component of
(c)
BAc
(d)
AxC
(Bx
Given
C)
vectors
=$2 -ý+23 and B
A
C
Given
vector parallel to
unit
3.8
By
(a)
expansion
The
A
starts
that A
is
orthogonal
is
9
$3 -22,
and whose
(a)
Show
The
vector
relation
A at point P
in Cartesian
for
the
=
(1,
C
find
relation
for
the
vector
Eq.(3.33).
Answer(s)
between the
lines
by
+2y
4.
triple
available
in
Appendix D.
triple
the
and ends at point P on the line such
Find an expression for A
direction is
line.
two
vectors
is given
C=
â(B
where â is
given
by
(b) The
product
given
D
defined
to
A
is
in the
a)=A(B A)
A2
the unit vector of
vector
component of B
by
-1,2).
product
A and B,
defined as the vector
A
direction of
coordinates, prove:
scalar
to
given
that,
perpendicular
The
described
at the origin
Eq. (3.29).
(b)
the smaller angle
find
is
A =$(x+2y)-(y+3z)+2(3x-y), determine
3.7
a
given line
Vector
3.14
whose magnitude
to both A and B.
perpendicular
a vector
to
C)
(Ax ý)-
3.6
A
3.13
B along C
$xB
(h)
vector algebra
x
Ax (Bx
(g)
2
at their intersection point.
(a) A andâ
(b)
Use
= -6.
= 8.
x +2y
3x +4y
1
perpendicular
Line
vectors
the following
plane are described by
x-y
lines in the
B.
Given
(e) A
Two
3.12
B.
between B, and Bz
line
+z =4.
2x
=
=å2-y3 +l
either direction of the
to
parallel
described by
and B.
In Cartesian coordinates, the three corners of a
triangle
(4, -4,4),and P3
(0,.4, 4), P2
(2,2, -4). Find
are Pi
of the triangle.
#a 3
=-5.
A
as the
vector
givenby
by
D =B
A(B
A)
A2
component of
B
A
certain plane
is
Coordinate
Sections 3-2 and 3-3:
described by
+42= 16.
2x +3y
Find the unit vector normal to the surface
from the origin.
in
the direction away
B =(z-3y) +ý(2x -3z) -(x
to B at point P
parallel
=(1, 0, -1).
unit vector
*3.17
Find
direction
is
A
3.18
G
a vector
whose magnitude
perpendicular
=*+ý2-2
and F
to both
A
at
vectors
=ý3-6.
is
E
described
is
Cartesian
y), find a
(b)
P=
d) P= (-2,2,-2)
(1, 1,3)
(c)
4 and whose
F,
where
the equation:
=x-1.
is orthogonal
the
to
at point
P2 on the
Find an expression
line.
for A.
Vector
E
E
field
is
given
by
Determine the component of E
surface
R
=2
P
point
at
3.20 When
sketching
a vector
we
of
to
(a) Pi
(2, 7/4, -3)
(b)
=
P2 =
(3,0,
c)
P = (4,7, 5)
3.24
(a)
=R 5R cos 0sin
+$3sin d.
cos
the coordinates of the following points
Cartesian coordinates:
Convert
-2)
the coordinates of the following points
from
coordinates:
cylindrical
Convert
spherical
3.19
to
P=(0,0,2)
3.23
P = (0, 2) and ends
at point
starts
which A
by
the coordinates of the following
noi
lowing points
from
and spherical coordinates:
cylindrical
(a) P=1,2,0)
+
and
Systems
Convert
cylindrical
given line
y
line,
3.22
Given
3.16
Vector
VECTO
ANALYSI
3.15
E
3
CHAPTER
172
to
=
=
P=
P
(5, 0, 0)
7)
(b) Pa
(5,0,
(c)
(3, 7/2, 0)
tangential to the spherical
=(2,30°,60°).
or demonstrating
often use arrows, as
the
spatial
variation
Fig. P3.20, wherein
the length of the arrow is made to be
proportional to the strength
of the field and the direction of the arrow is the same as that of
the
field's.
the vector
away from
field,
The
sketch
field
E
shown
=fr,
the origin
in
Fig. P3.20, which represents
in
consists of arrows
and
pointing
radially
their lengths increasing linearly
in
proportion to their distance away from the origin. Using this
arrow representation, sketch each of the following vector fields:
(a)
Ej=-iy
(b) E2
(c)
(d)
(e)
E3= åx +9y
=ix +y2y
Es =dr
E4
)E
3.21
=r
=fsin
o
Use arrowsto
Ej =îx ýy
E2 = -¢
Es =(1/x)
(d) E4 =fcos o
sketch each
of the following
vector
fields:
(a)
(b)
(c)
Figure P3.20 Arrow
(Problem
3.20).
representation
for vector field
E =tr
173
PROBLEMS
25
the
TIse
appropriate
area ds
a)r=3;
the
3.30
differential
,9
S
T/2
ST;
=T/4; -2
c)2<r<,
=f(cos d +3z) -¢(2r +4sin o) + (r-2)
B=-fsino + cos o
A
z ==0
find
<zs2
0
d) R 2:0s0sT/3;
<dsT
=
0
5;
T/3; 0 <o < 27
e) 0 R
Alsosketch
3.27
A
(6)
A
volumes described
o
05090,
of
a sphere
30
described
is
< 90°.
P
(c)
by
The enclosed volume.
Also
3.28
A
field
vector
is
0<R<2,
points:
= (2,7, 3)
(b)
described
(a)
(b)
3.29
point
The vector component
Transform
by
The vector component of
At
spherical
a
given point
coordinates
in
into
P
the surface of the cylinder
P, find:
of
E
E
=
to
(b)
given
spherical coordi-
=
Po
T)
(4, 7/2,
the vector
=R e
cos
sin?
cylindrical
(2, T/2,
=
d +8cos d
and
coordinates
sin
then
A
å(x
+y)
at Pi
=(1,2,3)
Ps=(1, -1,2)
in
=R4 +02-.
D
=R
(e) E
at P2
sin
6 +0cos 6
+
coso at
(a)
(C)
in the direction
B the direction of A.
The vector component of B
perpendicular to A.
of
in
A
B
(c) C
(b)
component
=Rcos +0sin o +
sin 0
at
Ps
=
(3,
7/2, 7)
Transform the following vectors into
spherical
nates and then evaluate them at the indicated
points:
Find:
(D) The vector
at
= (1,0,2)
3.35
B=-2 +$3.
of B
it
7/2).
B =(y -x) +ýx -y)
(d)
The scalar component, or projection,
of A.
o
evaluate
P=(2,7/2,7/4)
a)
pairs of
c)C=y/(x +y*)- ýx-/«* +y)+24
tangential to the cylinder.
A and B are
in
Transform the following vectors into cylindrical coordinates and then evaluate them at the indicated
points:
the cylinder.
by
A
cylindrical
3.34
(a)
perpendicular
space, vectors
in
and P2
(0, 2,3)
(2,7/3, 1) and P4= (4,7/2,3)
(1, 1,2)
P=
3.33
given in cylindrical coordinates
on
4)
(3,
() Ps = (3, 7, 7/2) and
section.
is located
byr =2. At
0) and
=
(a) Pi
E =fr coso +¢rsin o +z'
P
Cartesian
Determine the distance between the following
A
Point
(3,
P4
(4,
nates.
3.32
Find thefollowing:
(b)
of the
in
3)
(1,
/3,1)
(2, T
coordinates.
The surface area of the spherical section.
the outline
B at
and
the following pairs of points:
Find the distance between
Ps
(a)
sketch
A
both
to
vector perpendicular
unit
coordinates.
by the following:
the outline of each volume.
section
and
at (2, 7/2,0)
= 2, and P2 = (-2, -3,-2)
= 7/4,
6) =(1,7/4,3) and
P = 7,0)
Ps =
7/2,
ST; 0<z<2
(a)2srs5;
5; 0 s6 ST/3; 0<o < 27
b) 0s R
sketch
eAB
(a)
T/2
Also
(a)
3.31
the outline of each surface.
Find the
1 26
Given vectors
0O S 7/3; -2szs 2
2r S
b)
for
deter
urtace
Surtaces:
expression
termine the area of each of
the
following
to
y +ýxz
+24
+
y+z)-
= fcos o-¢
sin
Ps= (2,T/4,2)
(d)
at Pi
=(1,-1,2)
2(+ y)
o+
cos
at
osin d
P =(-1,0,2)
at
D =y/(r2+ y*)-a/a+y)+24
P4=(1, -1,2)
coordi-
at
CHAPTER3 VECTORANALY
NSS
174
Sections 3-4 to 3-7:
Gradient, Divergence,
and Curl
Operators
3.36
Pi=(0,3)
Find the gradient of the
following scalar functions:
(a)
T
(b)
V
(c)
U
3/(x +z)
=xy24
zcoso/(1 +r2)
=
P2=3,0)
W =eRsin e
S =4xe +y
(d)
*(e)
) N
g)
Figure P3.41
For
each
of the
analytical solution for
scalar
following
VT
and generate
a
obtain an
fields,
corresponding
arrow
3.42
For
= 10+x, -10 < x
(b)T =x,for -10 <x 10
T
for
its
10
evaluate
T =100+xy,for
-10<x< 10
d) T xy, for -10<x, y < 10
(e) T 20+x +y, for -10
<x,y < 10
(c)
1+sin(7x/3),for -10<x<
T
,
(g) T 1+cos(Tx/3),
T
(h)
3.43
=
rcos
15
)T15+r
cos
x
-10 <
for
for
evaluate
10
3.44
=0,
10 atz
3.39
Follow
find
its
A
For the
directional
=( -ýz)
3.41
0Srs
0<
27.
segment P
=
the range
=(5, 7/4,7/2).
sin
s
6, determine it
direction
V
A
in
the
R
and
then
vector
fields
is
displayed
in
form of
analytically
and
a vector representation.
then compare the result
expectations on the basis of the displayed
scalar
is given
7T
-îcos x
A =
sin
y
+ý
sinx cos
Determine
with
your
pattern.
y, for
by
=ie z
t
similar to that leading
given
by
function
derivative
along
and then evaluate
Evaluate
U
T(z).
a procedure
to derive the expression
coordinates.
3.40
P
along
dete
f and
then
function
scalar
,
direction
=(2, 7/4,3).
Each of the following
Fig. P3.44
10
The gradient of a scalar function
=
T
at
=zer/ cos
radial
d,for0
VT
If
P
derivative
it
T
10
(a)
3.38
at
For the
directional
function
derivative along the
directional
it
scalar
the
representation.
(f)
3.41.
M =R cosesino
3.37
(a)
Problem
=r cos
it
Eq. (3.83) for
to
V
=xy -z,
V
the
direction
Eq. (3.82)
in spherical
determine
of
= -1,4).
of E =*x -ýy
at
the line integral
to Pa of the circular path
P
vector
(1,
along the
shown
in Fig. P3.41.
Figure P3.44(a)
-7 <X, y <T
175
PROBLEMS
y
b)A=Ns
+ycos 2x,for
-7x,y<T
Figure P3.44(b)
(c)
-$xy
+ýy5,
for
-10
< x,y <
10
(e)
A
=r,for -10 <r<
)
A
10
10
Figure P3.44(e)
=åxy>,
for
-10< x, y
10
-10
-10
Figure P3.44(c)
d)A=-$cOs
x+ý
sin
y,
for
-7
X, y
Figure P3.44()
ST
()A
=Rxy2 +ýr*y,
for
-10<x, y <
T0
Figure P3,44(d)
Figure P3.44(g)
10
CHAPTER 3
176
VECTOR
ANALYSIS
h) A
i
sin
(0)+$sin().for -10 x,y
10
field
Vector
3.45
E
is
characterized by the
ne
;(b) the magnitu followin
points along
properties:
de of
the distance fromthe
function of only
origin; (c)
=12, everywhere. Find
Find an vanishesa
the origin; and (d)
E that satisfies these properties.
expression
(a)
E
for
For the vector
3.46
divergence
(a) The
each and
(b) The
A
=fr+or
cos d,for
rs2
10
r
V E
vector
field
units
over the cube's volume.
= rl0e
E
field
-23z,
veri
the cylindrical region
enclosed
for
=2, z =0, and z =4
3.48 A
the
by
D =ir° exists
the region between
in
=
two
r
surfaces defined by
andr=2.wi
and z
both cylinders extending between z
5. Verifv th
he
the
following:
divergence theorem by evaluating
concentric
cylindrical
=0
1
Figure P3.44(h)
of
theorem
divergence
verify
ofa
2
the vector
For
3.47
-ýyz-2rv
flowing through the sure
the origin and with sides equal
to the Cartesian axes.
at
parallel
integral
= Xrz
flux
outward
total
cube centered
E
field
by computing
theorem
Ei
E
VE
=
(a)D-ds
(b)VD dv
For
D =R3R-,
the vector field
theorem
the divergence
3.50 For the vector field
(a)
10
A =fr?+or2 sind,for0Srs 27.
o<
E dl
around
=
R
sphericalshells defined by
Figure P3.44(i)
E
the
and
R
triangular
sides
of
between
the
2.
=*ry-ý(x*+2y),
contour
calculate
shown
Fig. P3.50(a).
(a)
Figure
P3.50
Problem 3.51.
Figure P3.44)
evaluate both
for the region enclosed
I
3.49
(b)
Contours
for
(a) Problem
3.50 and
(b)
in
177
PROBLEMs
TxE)
ds over
the area
ofthe
triangle.
3.50 for the
Problem
Prob
contour
Repeat
shown
in
(0,3)
.51 sob)
eorem for the
Stokes's
vector
Verify
B
(Tr cos
field
Li
o+d
sin o)
-3,0) L3
the following:
by
evaluating
the
over
B-dl
semicircular
contour
shown
Figure P3.55
in
Problem
3.55.
P3.52(a).
Fig.
3.56
xB)-ds
overthe
surface
ofthe semicircle.
Determine
if
each
of the following
vector
fields
solenoidal, conservative, or both:
(a) A =ir2-ý2xy
i2-ýy2 +ž2
C= (Sin)/r2 +d (cos o)/r2
B
(b)
(c)
D
(d)
R/R
(e)E-(3-)+z
()F=Ry +ýx)/(r2 +y)
G
(g)
(a)
r2+z)-9y2 +x)-(y
H=R(Re-R)
(h)
(b)
Find the Laplacian of the following scalar functions:
3.57
P3.52
Fieure
Problem
53
Fig.
Contour paths for
3.52 for
Problem
V 4xy2z3
(b) V
xy
yz
(c) V 3/(x2
(a)
the
contour
shown
(d)
= + +zX
+y)
V =
cos
p
5e
(e)
V
in
=10e-Rsin
P3.52(b).
Stokes's
Verify
A=RCOS+¢ sin6
of unit
for
theorem
by
it
evaluating
the
on
vector
field
the hemisphere
theorem
Stokes's
Verify
+
Bde
circle,
3.58
Find the Laplacian of the following scalar functions:
(a) Vi
(b) Va
radius
B= (f cos
a)
Problem 3.52 and (b)
3.53
Repeat
3.54
3.55
(a)
as
(x
sin
o)by
for
Fig. P3.55,
B).ds over
vector
field
evaluating:
over the path comprising
shown in
the
+)
a quarter
section
of a
and
the surface of the quarter section.
=
2
=(2/R>)cos
10r3 sin
sin
o
is