M408M, Multivariable Calculus

This page is always under construction, so
you
should check it regularly.

The written assignments will appear on this page. At the beginning
of each discussion section, you will be given a written assignment
consisting of several interesting and challenging problems. You will
work in class on one or more of these problems, as directed by the TA,
during the discussion. You will continue to work outside of class on
these problems, and then your carefully written solutions will be
collected **at the beginning of discussion on Monday of the
following week**. Some of the problems will be graded. In order to
receive credit, you must put your name and unique number and time of
your discussion section at the top of the page and show all of your
work. Your exercises must be well-labeled, neat, and in order, and the
work must be stapled. Above all, your assignment must be turned in
before the discussion begins. There will be approximately a dozen such
weekly assignments; we will drop the lowest two.

The Quest assignments will appear on (you guessed it) Quest. In general, there will be a pre-class and a post-class Quest assignment for each class meeting. The pre-class assignment is due at midnight the night before the class. The post-class assignment is due at 6PM on the day before the subsequent class. In other words, the pre-class assignment for Tuesday is due Monday night, and the post-class is due Wednesday afternoon; the pre-class assignment for Thursday is due Wednesday night and the post-class is due on Monday afternoon. Written assignments are due on Monday afternoon in discussion section.

The class schedule, and hence the schedule of Quest assignments, can be found here.

**Homework #1**, due Monday, September 9

Stewart, page 642, problems 24, 25, 26, 37.

Stewart, page 651, problems 29, 32.

Stewart, pages 662-3, problems 16, 20, 26, 62.

**Homework #2**, due Monday, September 16

Section 10.4, page 668, problems 8, 26, 45, 46.

Section 10.5, page 676, problems 9, 16, 18, 51.

**Homework #3**, due Monday, September 23

Section 10.6, page 684, problems 5, 8, 12, 16, 29.

Section 12.1, page 790, problems 6, 10, 12, 16

Section 12.2, page 798, problems 2, 8, 30, 38

**Homework #4**, due Monday, September 30, can be found
here.

**Homework #5**, due Monday, October 7, can be found
here.

**Homework #6**, due Monday, October 14

Section 12.6, pages 832-4, problems 10, 21-28, 49

Section 13.1, pages problems 15, 16, 21-26

Section 13.2, problem 52.

There's only one written problem from Section 13.2. None of the
other ones in the book looked very instructive.

**Homework #7**, due Monday, October 21

Section 13.3, problems 16, 38 and 56.

Section 13.4, problems 2, 6, 16, 36, 38, 42.

**Homework #8**, due Monday, October 28

Section 14.1, problems 5, 7, 8, 38

Section 14.2, problems 4, 14

Section 14.3, problems 4, 6, 10, 96.

**Homework #9**, due Monday, November 4

Section 14.4, problems 12, 22, 23, 32, 33, 42

(Just one section because of the midterm.)

**Homework #10**, due Monday, November 11

Section 14.5, problems 36, 44

Download a map of Enchanted Rock State Natural Area at
http://www.tpwd.state.tx.us/publications/pwdpubs/media/park_maps/pwd_mp_p4507_119c.pdf and print out 3 copies. On one copy, sketch the gradient of the
height function.
On a second copy, mark points where the gradient is unusually large. How
would you describe the terrain near those points? On the third copy, mark
points where the gradient is practically zero. What is happening near those
points? Which ones are local maxima? Local minima? Saddles?

Section 14.6, problems 32, 34, 44

**Homework #11**, due Monday, November 18

Problem 1) Suppose we have a line $y=mx+b$ that we are using to find a
linear fit to the points $P(1,1)$, $Q(2,2)$ and $R(3,4)$ in the
$x$-$y$ plane. Let $\epsilon_1$ be the amount that $P$ lies above the
line (which may be negative if $P$ is actually below the line),
$\epsilon_2$ be the amount the $Q$ lies above the line, and
$\epsilon_3$ be the amount that $R$ lies above the line.

(a) Write formulas for $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ as
functions of $m$ and $b$.

(b) Let $E(m,b) = \epsilon_1^2 + \epsilon_2^2 + \epsilon_3^2$ be
the total squared error. We want to find the value of $(m,b)$ that
minimizes $E$. (Remember that $E$ is a function of $m$ and $b$, not a
function of $x$ and $y$!) This involves taking a gradient and setting
it equal to zero. Write down the gradient $\nabla E(m,b)$. Simplify as
much as possible.

(c) Now solve the equations $\nabla E(m,b) = 0$ to figure out what the
best values of $m$ and $b$ are. So what is the equation for the best
line through the points $P$, $Q$, and $R$?

(d) Using the best
line you found in (c), compute $\epsilon_1$, $\epsilon_2$ and
$\epsilon_3$. Does you consider this line a good fit or a bad fit?
Explain.

Problem 2) Find the extremal values of $f(x,y,z) = 2xy + 3z^2$ on the
surface $x^2+y^2+z^2=1$. You don't need to know any linear algebra to
do this, but what you are doing is equivalent to finding the
eigenvalues and eigenvectors of the matrix $\begin{pmatrix} 0&1&0 \cr
1&0&0 \cr 0&0&3 \end{pmatrix}$.

Stewart, Section 14.7, problems 4, 16, 40, 50

Stewart, Section 14.8, problems 14, 30, 40

**Homework #12**, due Monday, **December 2**

Note that there isn't any homework due Monday, November 25, mostly because of
the exam. However, I recommend that you do as much of this problem set as
possible by Nov 25, since you probably don't want to spend Thanksgiving
weekend doing calculus.

Stewart Section 15.2, problems 6, 8, 12, 38

Stewart Section 15.3, problems 26, 50, 54, 56.

Stewart Section 15.4, problems 24, 28a (skip b), 36