HOMEWORK 8 SOLUTIONS
ROBERT MCGUIGAN
MATH 31CH
1. By Greens theorem
(3 + 2y 2 ) (x2 + 3y 2 1) dxdy
(x2 y + y 3 y )dx + (3x + 2y 2 x)dy =
D
C
4 x2 y 2 dxdy,
=
D
where D is the region bounded by C .
The integrand is positive everywhere inside the disk of ra

HOMEWORK 6 SOLUTIONS
ROBERT MCGUIGAN
MATH 31CH
1. (i) By denition, we have
F = F1 dx1 F2 dx2 .
(ii) For the eld in question we have
F = dx dy.
Note that any directed line segment of length 1 can parametrized in the following form:
: [0, 1] R2 , (t) =
x0

HOMEWORK 7 SOLUTIONS
ROBERT MCGUIGAN
MATH 31CH
6.6.8. Let f, g : R4 R be dened by:
x1
x1
x2
x2
f = x2 + x2 + x3 x4 , g = 1 x4 .
1
2
3
x3
x3
x4
x4
Then M is given by f = 0, and X is given by f = 0, g 0. Note that X is closed (it is the
intersection of

HOMEWORK 5 SOLUTIONS
ROBERT MCGUIGAN
MATH 31CH
1. We parametrize the four sides of the square separately:
1 : [0, 1] R2 , (t) =
t
.
0
Then we have:
1
= t.
0
(xdx + ydy ) P t
0
Then
1
xdx + ydy =
[1 ([0,1])]
0
2 : [0, 1] R2 , (t) =
1
tdt = .
2
1
.
t
Then w

HOMEWORK 3 SOLUTIONS
ROBERT MCGUIGAN
MATH 31CH
Problem 4.10.8
(b) We use the change of variables
y
u = xy, v =
x
We can rewrite the inequalities dening the region as
1 u a, 1 v b.
We have
dxdy =
Instead of computing the Jacobian
change (which is easier)

RECOMMENDED REVIEW PROBLEMS FOR FINAL EXAM
First, review all the homework problems. In addition, you may want to solve the following
problems. Solutions will not be provided, but to make up for it, I will hold oce hours during
exam week to answer any ques

HOMEWORK 2 SOLUTIONS
ROBERT MCGUIGAN
MATH 31CH
1. Let A be an n n matrix. We have seen in class that A is invertible if and only if det A = 0.
On the other hand, the determinant is the product of eigenvalues. Thus det A = 0 if and only if A
doesnt admit 0

RECOMMENDED REVIEW PROBLEMS FOR MIDTERM II
1.C
(i) We calculate the derivative of the two equations to obtain the matrix
2x z 2y
x
2w
.
yz w xz 2z xy x + 2w
At (1, 0, 0, 1) this matrix becomes
2 0 1 2
1 0 0 1.
This is clearly surjective since the columns

Math 31CH - Spring 2013 - Midterm I Solutions
Problem 1.
Calculate the mass of a plate with dentsity (x, y ) = x contained between the curves
x2 y = 1, x2 y = 2, x = y, x = 2y.
Solution:The mass of the plate is
x dx dy.
dx dy =
mass =
D
D
We change varia

REVIEW PROBLEMS FOR MIDTERM I
First, review all the homework problems. In addition, you may want to solve the following
problems. Solutions for the textbook problems can be found in the solution manual.
(1) Linear algebra. Determinants. Volumes of paralle

SOLUTIONS
Solutions for the textbook problems can be found in the solution manual.
1.A We are told that A 2I is not invertible, hence = 2 must be an eigenvalue. Eigenvalues
are roots of the characteristic polynomial hence
A (2) = 0 = 23 3 22 + 2a = 0 = a

RECOMMENDED REVIEW PROBLEMS FOR MIDTERM II
First, review all the homework problems. In addition, you may want to solve the following
problems. Solutions for the textbook problems can be found in the solution manual.
(1) Orientations.
A. Solve problem 6.3.

Math 31CH - Homework 8. Due Friday, June 7.
Instructions: To help you keep up, the day on which you will have the needed background for each
problem is written next to it.
1. (Monday.) For what simple closed curve C does the line integral
(x2 y + y 3 y )

Math 31CH - Spring 2013 - Midterm II Solutions
Problem 1.
Consider the surface S R4 given by the equations:
x2 + yz yw = 1, xy + zw + xw = 3.
Orient S using the dierential form
= dz dw.
Write down a positive basis for the tangent space to S at the point

Math 31CH - Spring 2013 - Midterm II
Name:
Student ID:
Instructions:
Please print your name and student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanatio

Math 31CH - Spring 2013 - Midterm I
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation wi

Math 31CH - Homework 6. Due Friday, May 17.
Instructions: To help you keep up, the day on which you will have the needed background for each
problem is written next to it.
1. (Monday.)
(i) Write out the explicit denition of the ux form of a vector eld in

Math 31CH - Homework 3. Due Friday, April 19.
1. [Volumes of parallelograms.] Solve problem 5.1.1 from the textbook.
2. [Change of variables.] Solve problems 4.10.8, 4.10.13, 4.10.14, 4.10.20 from the textbook.
3. [Change of variables.] Evaluate
D
(2x + y

Math 31CH - Spring 2013 - Final Exam
Problem 1.
Show that for any constants 0 < a < b and 0 < c < d, the area of the plane region bounded by
the curves
x2 = ay, x2 = by, x3 = cy 2 , x3 = dy 2
equals
15
(b a5 )(c3 d3 ).
15
Solution: The area equals
dx dy.

Math 31CH - Spring 2013 - Final Exam
Name:
Student ID:
Instructions:
Please print your name and student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work.
There are 11 questions whi