TEST4/MAC2313
Page 4 of 6
______________________________________________________________________
7.
(10 pts.) Compute the work in moving a particle along of the path in
the xyplane that goes from the point (1,1) to the point (1,1) along the
curve
y
=
x
5
against the force field defined by
F
(
x
,
y
)
y
x
,
yx
3
.
The Toidi’s parameterization for the curve, in the correct direction,
is given in a vector form by
r
(
t
) = <
t
,
t
5
> for
t
ε
[1,1]. Consequently,
r
′
(
t
) = < 1 , 5
t
4
> and thus,
w
⌡
⌠
C
F
d
r
⌡
⌠
C
F
(
r
(
t
))
r
(
t
)
dt
⌡
⌠
1
1
(
t
5
t
)(1)
(
t
5
t
3
)(5
t
4
)
dt
⌡
⌠
1
1
t
5
t
5
t
12
dt
10
⌡
⌠
1
0
t
12
dt
10
13
.
How did Em Toidi know that a parameterization is needed?? First, the field
is NOT CONSERVATIVE. [Go check this, Frodo.] This means that the
Fundamental Theorem of Line Integrals CANNOT BE USED. Second, the curve is
NOT CLOSED, although the curve is simple. As a consequence, Green’s
Theorem CANNOT BE USED. This means finally, you are stuck with the
"definition." A nice, easy parameterization is a must. This is actually
an easy path integral, oddly. Or was it evenly????? [E.T. used both!!]
______________________________________________________________________
8.
(10 pts.) Starting at the point (0,0), a particle goes along the
yaxis until it reaches the point (0,4). It then goes from (0,4) to (4,0)
along the circle with equation
x
2
+
y
2
= 16. Finally the particle returns
to the origin by travelling along the xaxis.
Use Green’s Theorem to
compute the work done on the particle by the force field defined by
F
(
x
,
y
) = < 5
y
3
, 5
x
3
> for (
x
,
y
)
ε
2
. [Draw a picture. This is easy??]
W
C
F
d
r
C
5
y
3
dx
5
x
3
dy
⌡
⌠
⌡
⌠
R
∂
∂
x
(5
x
3
)
∂
∂
y
(
5
y
3
)
dA
15
⌡
⌠
⌡
⌠
R
x
2
y
2
dA
15
⌡
⌠
π
π
/2
⌡
⌠
4
0
r
3
drd
θ
480
π
. [
Corrected
.]
Picture: Quarter disk of radius 4 in the second quadrant bounded by the y
axis on the east and the xaxis on the south. Trace the boundary counter
clockwise.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
TEST4/MAC2313
Page 5 of 6
______________________________________________________________________
9.
(10 pts.) (a) Show that the vector field
F
(
x
,
y
)
< cos(
x
)
e
y
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '06
 GRANTCHAROV
 Multivariable Calculus, ρ, 10 pts, π

Click to edit the document details