Name
Math 311
Exam 3 Version A
Section 502
Spring 2015
Solutions
P. Yasskin
1
/20
3
/30
2
/36
4
/26
Total
/112
Points indicated. Show all work.
1.
20 points
Compute
F dS for F
y, x, z
S
over the "clam shell" surface, S, parametrized by
R r,
for r
r cos , r sin , r sin 5
3 oriented upward.
HINTS: Use Stokes Theorem.
What is the value of r on the boundary?
Stokes Theorem says
3 on the boundary, the boundary curve is r
The velocity is v
9 sin 2
9 cos 2
45 sin 5
F ds
y, x, z
cos 5
2
F dS
R 3,
3 cos , 3 sin , 3 sin 5
3 sin , 3 cos , 15 cos 5
The vector field on the boundary is F r
F v
S is the boundary curve.
S
S
Since r
F ds where
F dS
9
cos 5
2
F vd
0
45 sin 5
3 sin , 3 cos , 3 sin 5
9
45 sin 5
cos 5
0
d
9
9 sin 2 5
2
2
18
0
This can also be done directly without using Stokes Theorem.
1
2.
36 points
basis
e1
Span e 2x
Let V
e 2x
e
2x
,
e
2x
, e 2x
e 2x
e2
2x
e
be the vector space of functions spanned by the
2x
e
df
Consider the linear operator L : V
V given by L f
3 . Our goals are to compute the
dx
eigenvalues and eigenfunctions of the linear operator L, to find the similarity transformation which
diagonalizes the matrix of L and use this similarity transformation to compute a matrix power.
a.
5 pts
Find the matrix of L relative to the
e1, e2
basis. Call it
A.
e e
b.
d e 2x
e
2x
L e1
L e 2x
e
2x
L e2
L e 2x
e
2x
3 pts
Find the characteristic polynomial for
3
3
dx
d e 2x
e
6e 2x
6e
2x
6e 2x
6e
2x
6e 2
A
2x
e e
dx
6e 1
0 6
6 0
A.
e e
Factor it and identify the eigenvalues of
A . These are also the eigenvalues of L.
e e
det A
c.
6
1
8 pts
2
6
36
6
Find the eigenvector(s) of
6
6, 6
for each eigenvalue, as vectors in
A
2
.
e e
Name them v 1 and v 2 .
6:
6:
d.
6 pts
6 6
0
1 1
0
6 6
0
0 0
0
v1
6
6
0
1
1
0
6
6
0
0
0
0
Convert the eigenvectors of
A
v2
a
r
b
r
a
r
b
r
v1
v2
1
1
1
1
into eigenfunctions of L as functions in V.
e e
Name them f 1 and f 2 and simplify them.
Then compute L f 1 and L f 2 to verify f 1 and f 2 are eigenfunctions.
Hint: Remember that the components of v 1 and v 2 are components of f 1 and f 2 relative
to the e 1 , e 2 basis.
f1
f2
L f1
L f 21
1e 1
1e 1
1e 2
e 2x
e
2x
e 2x
e
2x
2e
2x
1e 2
e 2x e 2x
e 2x e 2x
2e 2x
d 2e 2x
3
12e 2x
6 2e 2x
6f 1
dx
2x
d 2e
3
12e 2x 6 2e 2x
6f 2
dx
2
e.
3 pts
Using the eigenfunctions as a new
to the f 1 , f 2 basis. Call it D .
f
f1, f2
basis for V, find the matrix of L relative
f
Since
f1, f2
is a basis of eigenvectors, the matrix of L relative
to the
f1, f2
basis will be diagonal and the diagonal entries will
6 0
D
f
0
f
6
be the eigenvalues.
f.
5 pts
Find the change of basis matrices
and
C
f1
g.
1e 1
1e 1
1e 2
1
1 1
C
1e 2
1
e f
Since
A
C
f
D
e e e f
basis to the
C
1
f
C
e
e f
1
2
f
f
1
1
1
1
2
1
2
1
2
1
2
S 1 D S.
e
we identify S
C
f
1
e
C.
f
e
Compute A 12 and A 25 .
4 pts
6 0
With D
D 12
D 25
e1, e2
e
2 pts
A and D are related by a similarity transformation A
Identify S as C or C .
e f
h.
f
bases. Be sure to identify which is which.
f1, f2
f2
between the
C
e f
6 25
0
6
6 12
0
0
6 12
1 0
0
1
6
1 0
0
1
we have
6 12 1 and A 12
and A 25
S 1 DS
S 1 DS
25
6 24 S 1 DS
12
S 1 D 12 S
S 1 D 25 S
6 24 A
6 24
6 25 S
6 12 S 1 1S
1
6 12 1
1 0
0
6 12
0
0
6 12
S
1
0 6
0
6 25
6 0
6 25
0
3
3.
30 points
The density, , of an ideal gas is related to its pressure, P, and its absolute
P where k is a constant which depends on the
temperature, T, by the equation
kT
particular ideal gas. We are considering an ideal gas for which k 10 4 atm m 3 /kg/°K. At
2, 1, 3 m and has
the current time, t t 0 , a flying robotic nanobot is located at x, y, z
velocity v
. 4, . 5, . 2 m/sec. The nanobot measures the current pressure is P 2 atm
while its gradient is P
. 03, . 01, . 02 atm/m. Similarly, the nanobot measures the current
temperature is T 250 °K while its gradient is T
3, 2, 4 °K/m.
a.
2 pts
P
kT
b.
6 pts
T
2 atm
10
4
80 kg/m 3
3
atm m /kg/°K
250 °K
Find the Jacobian matrix of the density
4 pts
P
,
D P, T
D x, y, z
Find the Jacobian matrix
4 pts
D x, y, z
Dt
P
x
T
x
D
D P, T
in general (in terms of symbols like
P
y
T
y
D P, T
D x, y, z
1
10
t t0
4
t0.
,
250 10
4
2.
250
40, . 32
2
P ) and
y
in general (in terms of symbols like
t0.
P
z
T
z
P
D ,T
D x, y, z
T
Find the Jacobian matrix
dx
dt
dy
dt
dz
dt
D
D ,T
1 , P
kT kT 2
T
then at the current time t
d.
.
), then in terms of P and T, and finally at the current time t
D
D P, T
c.
Find the current density,
D x, y, z
Dt
. 03 . 01 . 02
t t0
2
3
in general and then at t
4
t0.
.4
v
D x, y, z
Dt
v t0
t t0
.5
.2
4
e.
6 pts
time t
Find the time rate of change of the pressure as seen by the nanobot, at the current
t 0 . Is the pressure currently increasing or decreasing?
.4
dP
dt
P
t t0
t t0
v t0
. 03, . 01, . 02
.5
. 012 . 005 . 004
. 003
.2
The pressure is decreasing.
f.
8 pts
time t
Find the time rate of change of the density as seen by the nanobot, at the current
t 0 . Is the density currently increasing or decreasing?
d
dt
D
D P, T
t t0
40, . 32
t t0
D P, T
D x, y, z
t t0
. 03 . 01 . 02
3
2
4
D x, y, z
Dt
t t0
.4
.5
.2
40, . 32
. 003
.6
. 12 . 192 . 072
The density is increasing.
5
26 points
4.
Compute the integral
y4
x dA over the
region in the first quadrant bounded by
y
1
2
x , y
2
x , y
3
2
2
x , and y
4
2
x .
5
0
0
a.
4 pts
Define the curvilinear coordinates u and v by y
What are the 4 boundaries in terms of u and v?
u
1
u
4 pts
b.
3
v
4
y
x2
u
r u, v
x2
v
x2.
2x 2
u
x2
u
2x 2
,u
v
v
y
u
x
u
1
4 2
u
v
2
v u
2
v
2
Find the coordinate tangent vectors:
v
ev
r
v
1
2 2
v
1
,1
u 2
1
u
,1
2
Compute the Jacobian factor:
1
2 2
1
2 2
x, y
u, v
x, y
u, v
6 pts
v
v
2
1
2 2
J
x2
v
r
u
8 pts
v
5
u
u
eu
d.
e.
2y
x u, v , y u, v
4 pts
c.
x 2 and y
2
x
Solve for x and y in terms of u and v. Express the results as a position vector.
Add and subtract:
y
v
u
1
1
u
1
2
v
u
1
2
1
2 2
v
v
1
1
4 2
1
v
1
v
u
1
2 2
1
v
u
1
u
Compute the integral:
5
3
x dA
4
1
v
u
2
1
2 2
5
1
v
u
3
du dv
4
1
1 du dv
4
1 5
4
4 3
1
1
2
6