Math 31CH - Homework 5. Due Friday, May 10.
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. (Friday.) Integrate the form
= x dx + y dy
over the edges of the square with op
Math 31CH Homework 4 Solutions
May 3, 2014
Problem 5.3.8
The paraboloid:
z = x2 + y 2
can be parameterized as:
(u, v) = (u, v, u2 + v 2 ).
We want to integrate over all values where z 1, which means we want to integrate over the region:
C = cfw_(u, v)|u2
Homework 1 Solutions
Problem 4.1.6
Part A
We can use induction to show that
n
i2 = n(n + 1)(2n + 1)/6.
i=0
First we verify that the above identity is true for n = 0 and n = 1.
For the inductive step, assume it holds true for some n 1 and we prove it for n
HOMEWORK 2 SOLUTIONS
4.5.6. (a) Note that the smallest n-dimensional cube containing the n-dimensional unit sphere is
the n-fold product
[1, 1] [1, 1]
which has volume 2n . We need to show that the sequence
to proving that for n 2, we have
n
2n
is decreas
HOMEWORK 3 SOLUTIONS
Problem 4.10.8 (b) We use the change of variables
u = xy, v =
y
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)
(x,y)
(u,v)
(x, y)
dudv.
(u
HOMEWORK 6 SOLUTIONS
6.3.4. To orient a curve C in R2 we can use a normal eld to the curve, for instance the eld given
by the gradient of the equation
x + x2 + y 2 = 2.
This gradient eld is
1 + 2x
.
2y
6.3.6. To orient the surface, we can use the gradient
Math 31CH - Homework 1. Due April 7.
Riemann sums. Integrable functions. Sets of volume zero. Sets of measure zero.
1. (Monday-Wednesday.) Solve problems 4.1.5 and 4.1.6(a)(b). For problem 4.1.6(a) you may
want to use induction to prove that
n(n + 1)(2n +
Math 31CH - Homework 3. Due Monday, April 21.
Part I: Change of variables in two and three dimensions.
1. Solve problem 4.9.4 and 4.9.5 from the textbook.
2. Solve problems 4.10.8, 4.10.14, 4.10.20 from the textbook.
3. Evaluate
(2x + y 3)2
dx dy
2
D (2y
Math 31CH - Spring 2014 - Midterm I Solutions
Problem 1.
Using change of variables, nd the area of the rst quadrant region bounded by the curves
xy = 2, xy = 4, xy 3 = 1, xy 3 = 2.
Solution:We make the change of variables
u = xy, v = xy 3
so that
2 u 4, 1
Math 31CH - Spring 2014 - Midterm II
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 w
Math 31CH - Homework 2. Due Friday, April 12.
1. Show that a square n n matrix is invertible if and only if 0 is not an eigenvalue.
2. Using the characteristic polynomial, nd the eigenvalues of the following matrices. Are these
matrices diagonalizable?
1
Math 31CH - Homework 4. Due Friday, May 2.
1. Solve problem 5.1.1.
2. Solve problem 5.3.1.
3. Solve problem 5.3.8.
4. Read example 5.3.11. Use the same method to solve problem 5.3.10 and 5.3.21.
5. Solve problem 5.3.16.
6. Read example 5.3.10. Use the sam
Math 31CH - Homework 5. Due Friday, May 16.
Part I: Orientations.
1. Section 6.3: solve problem 6.3.7 and 6.3.11(b).
2. Section 6.3: solve problems 6.3.4, 6.3.6, 6.3.12.
3. Review Exercises for Chapter 6: solve problems 6.5, 6.6, 6.7(a).
4. Consider the s
Math 31CH - Last Homework. Due Friday, June 6.
This homework is longer and more dicult than the rest. To help you keep up, the topic and the
day on which you will have the needed background for each problem is written next to it.
0. Extra credit. (Boundar
HOMEWORK 7 SOLUTIONS
1. (i) By denition, we have
F = F1 dx2 F2 dx1 .
(ii) We have to integrate the form
F = x2 dy xy dx
over the parabola. We have the orientation-preserving parametrization:
t
.
2t2
: [0, 1] R2 , (t) =
Then
F P
t
2t2
1
= t2 (4t) t 2t2 1
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) Integration in rectangular coordinates. Fubinis
Math 31CH - Spring 2014 - Midterm II
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 w
SOLUTIONS
Solutions for the textbook problems can be found in the solution manual.
1.A We change the order of integration over the region
0 y x 1.
We nd
0 y x2 , 0 x 1.
and
x2
1
1
3
ex dy dx =
0
0
0
1 3
1
3
x2 ex dx = ex |x=1 = (e 1).
x=0
3
3
1.B The inte
Math 31CH - Homework 7. Due Wednesday, May 28.
Part I: Work and ux forms.
1.
(i) Write out the explicit denition of the ux form of a vector eld in R2 given as
F = F1 i + F2 j.
(ii) Find the ux of the eld
F = x2 i + xy j
through the parabola y = 2x2 orient