MA/PH 607
Howell
11/7/2011
Homework Handout IX
A.
For the following problems, we will use Cartesian
coordinates to describe points in the plane, and will
let
,
,
and
be the four curves from
V
V
V
V
"
#
$
%
lparen1
rparen1
lparen1
rparen1
!ß!
#ß%
to
indicated in the figure.
Note that:
+Ñ
\
V
V
"
%
and
consist of portions of the
–
and
]
–axes,
,Ñ
V
$
is a straight line, and
Ñ
BßC
C œB
each point
in
satisfies
.
lparen1
rparen1
V
#
#
In addition, assume
and
are, respectively, a scalar field and a vector field with
9
F
coordinate formulas
9
lparen1
rparen1
lparen1
rparen1
ˆ
‰
BßC œ B C
BßC œ #BC mathplus BmathminusC
#
#
and
.
F
i
j
1.
Find parametrizations for each of these four curves (the parametrizations for
and
V
V
"
%
can each be in “two pieces”).
2.
r
Determine
“
”
for each of the parametrizations just found, and then compute
.
(
V
5
F
†.
5 œ"ß #ß $ß
%
r
for
and
.
3.
Determine
“
”
for each of the parametrizations just found, and then
.=
set up
(computing if practical)
(
V
5
9
.=
5 œ"ß #ß $ß
%
for
and
.
4.
Using your results from the above, compute
(
I
F
†.
r
where
a.
(I.e, take
from
to
, then go backwards on
to
.)
I
V
V
V
V
œ
mathminus
!ß!
#ß%
!ß!
#
$
#
$
lparen1
rparen1 lparen1
rparen1
lparen1
rparen1
b.
I
V
V
œ
mathminus
"
%
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Homework Handout IX
page 2
B.
For these problems, we are using cylindrical coordinates
in three
e
f
lparen1
rparen1
3 9
ß ß D
dimensional Euclidean space.
1.
Suppose we have a curve
in the plane with a cylindrical coordinate parametrization
V
ß
r
lparen1 rparen1
lparen1
rparen1
lparen1 rparen1
lparen1 rparen1
> µ
> ß
> ß DÐ>Ñ
3
9
.
Determine the formulas for
and
in terms of
,
,
,
,
and
.
.
.=
r
.
.
.D
.>
.>
.>
3
9
e
e
k
3
9
2.
Assume that, in fact, our curve
is given by
V
r
lparen1 rparen1
lparen1
rparen1
lparen1
rparen1
lparen1 rparen1
lparen1 rparen1
> µ
> ß
> ß DÐ>Ñ œ $ß # >ß >
!lessthan>lessthan&
3
9
1
for
.
a.
Sketch the curve.
b.
Compute
where
the square of the distance from the origin to
.
'
V
9
9
.=
Ð Ñ œ
B
B
c.
r
Compute
where
is the vector field with cylindrical coordinate formula
'
V
F
F
†.
F
e
e
k
lparen1
rparen1
<ß ßD œ
Ð Ñ mathplus D
mathplus
9
9
cos
3
9
3
.
3.
r
r
Compute
and
when
is the semicircle paramtrized by
'
'
V
V
9
e
e
<
‚.
‚.
V
r
lparen1 rparen1
lparen1
rparen1
lparen1
rparen1
> µ
ß ß D œ $ß ß &
!lessthan lessthan
3 9
9
9
1
for
.
C.
r
Determine the formulas for
and
in terms of the standard spherical coordinate
.
.=
system
e
f
lparen1
rparen1
<ß ß
Þ
) 9
D.
r
Determine the formulas for
and
in terms of the parabolic coordinate system
.
.=
e
f
lparen1
rparen1
?ß@
from problem
in
.
K
Homework Handout VI
E.
Arfken & Weber, page 59 : 1 (note:
some positive constant), 2, 3
(These three
5 œ
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 Summer '11
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 Math, Cartesian Coordinate System, Coordinate system, Polar coordinate system, Coordinate systems, Homework Handout

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