Name
ID
MATH 311
Final Exam
Section 502
Fall 2000
1-5
/25 7
/11
6
/45 8
/24
P. Yasskin
>Ÿ4x " 3y dx Ÿ3x " 2y dy
with vertices
and
Ÿ0, 0 ,
Ÿ0, 3 Ÿ2, 0 .
(HINT: Use Green’s Theorem.)
a. 3
b. 6
c. 9
d. 12
e. 18
1. (5 points) Compute
counterclockwise around the edge of the triangle
2. (5 points) Let MŸ2, 2 be the vector space of 2 • 2 matrices. Consider the linear map
L : MŸ2, 2 ¯ MŸ2, 2 given by LŸX X X T
where X T is the transpose of X. Which of the following is FALSE or are they all TRUE?
a. DomainŸL MŸ2, 2 b. CodomainŸL MŸ2, 2 c. KernelŸL antisymmetric 2 • 2 matrices
d. ImageŸL symmetric 2 • 2 matrices
e. All of the above are TRUE
3. (5 points) If a jet flies around the world from East to West, directly above the equator, in what
direction does the unit binormal
B
point?
(HINT: B T • N )
a. North
b. South
c. East
d. Up (away from the center of the earth)
e. Down (toward the center of the earth)
1
4. (5 points) Let P 2 be the vector space of polynomials of degree at most 2. Consider the inner product
1
˜p, q ™ Find the angle between the polynomials rŸx 4 2
a. cos "1
5 3
"1 16
b. cos
15
5 3
"1
c. cos
4 2
"1 15
d. cos
16
4 3
e. cos "1
5 2
5. (5 points) Compute ;; F
dS for F
Ÿx
; 3x pŸx qŸx dx
0
1 " x 2 and sŸx x.
3, 3, 2
y x z y 2 z over the complete surface of the cylinder
C
x 2 y 2 t 4 and 0 t z t 3.
(HINT: Use Gauss’ Theorem.)
a. 24=
b. 48=
c. 96=
d. 144=
e. 324=
2
6. (45 points) Consider the vector space V spanned by
e1
cosh 2 x,
e2
sinh 2 x and e 3
cosh x sinh x.
USEFUL FACTS:
sinh 0
0
cosh 0 1
d sinh x cosh x
dx
d cosh x sinh x
dx
cosh 2 x " sinh 2 x
1
sinhŸ2x 2 sinh x cosh x
coshŸ2x cosh 2 x sinh 2 x
d sinh 2 x 2 cosh x sinh x
dx
d cosh 2 x 2 cosh x sinh x
dx
d cosh x sinh x cosh 2 x sinh 2 x
dx
a. (5 pts Extra Credit) Show e 1 , e 2 and e 3 are linearly independent.
b. (10 pts) Another basisfor V is
Find the change
f 1 1, f 2 coshŸ2x and f 3 sinhŸ2x .
of basis matrices from the f-basis to the e-basis, C and from the e-basis to the f-basis, C .
esf
f se
3
c. (5 pts) Consider the linear operator of differentiation
D:V
¯
V given by DŸg dg
dx
Since
D Ÿf 1 D Ÿ1 D Ÿf 2 DŸcoshŸ2x 2 sinhŸ2x 2e 3
D Ÿf 3 DŸsinhŸ2x 2 coshŸ2x 2e 2
0
0 0 0
the matrix of D relative to the f-basis is B
f sf
0 0 2
.
0 2 0
Find the matrix of D relative to the e-basis. Call it A .
ese
d. (5 pts) Find the matrix of D relative to the e-basis by a second method.
4
0 0 0
e. (16 pts) Find the eigenvalues and eigenvectors of the matrix B
0 0 2
.
0 2 0
f. (4 pts) Find the eigenvalues and eigenfunctions of the operator D.
5
7.
(11 points) Compute ;; 1x e
xy
dx dy over the diamond shaped
10
region between the curves
1
and
x
You must use the curvilinear coordinates
and
x uv
y uv.
y
x,
y
9x,
y
y
4.
x
5
0
1
2
3
a. (3 pts) Find the Jacobian:
b. (4 pts) Express each boundary curve in terms of u and v:
c. (2 pts) Express the integrand in terms of u and v:
d. (2 pts) Compute the integral:
6
8.
(24 points) Use two methods to compute
• F dS
;; S
for
F
Ÿy, "x, z sphere
2
x2
y2
over the piece of the
z2
25
for
0tzt4
with normal pointing away from the z-axis.
• F and compute the double integral ;; • F dS
a. (12 pts) Parametrize the surface, compute S
directly.
7
• F dS
b. (12 pts) By Stokes’ Theorem ;; S
where S is the boundary of S.
ds
>F
S
for each circle.
ds
Parametrize the upper and lower circles and compute > F
Be sure to discuss the orientation of the circles when you add up the integrals.
8