**Math
204, Spring 2016 **

**Honors
Mathematics II **

**Instructor
**John E. M^{c}Carthy

**Class **
MTuThF
11.00-12.00 in 207 Cupples
I on Monday, in 199 Cupples I on Tuesday,Thursday, Friday

**Discussion
Section ** F 2.00-3.00 in Room 199, Cupples I

**JM Office **
105
Cupples I

**JM Office Hours**
M 12.00-1.00, Tu
10.00-11.00, Th
12.00-1.00, and by appointment

**Phone **
935-6753

**Teaching
Assistant** **
**Tokio
Sasaki

**TS
Office Hours
**M 3.00-4.00, W** **12.00-1.00, Th 3.00-4.00, F 3.00-4.00 in Room 6, Cupples I

**Exams** There
will be three exams in the course:

1) Exam 1 In class.
Friday February 19.

2) Exam 2
Take Home. Due
Monday April 2.

3) Exam 3
Final exam. Tuesday May 10,
10.30-12.30.

Homework

There will be weekly homework sets during the semester,
assigned on
Tuesday and due the following Tuesday.

** Homework
1 **due January 26.

**Prerequisites**

Math 203, or permission of instructor.

**Description**

This is the second half of a one-year calculus sequence for
first year
students with a strong interest in mathematics.

The course will be challenging, with an emphasis on rigor and proofs.

If you complete the year-long sequence, you will not only cover all of
Math
233, but most of Math
318 and Math 310, and some
of Math 309.

More importantly, you will have learned what real mathematics is!

In the first semester, we covered the basics of proofs
(including
addressing why proofs are important, and not just a formal exercise),

revisited one variable calculus, and spent some time on vectors.

In
the second semester, we will cover matrices, functions of several
variables, partial and total derivatives, multiple integrals,

line and surface integrals, Green's, Stokes's and Gauss's theorems.

**Content**

Here is a __very__
tentative schedule. We will not stick
to it closely.

Week 1: Linear transformations and matrices. Determinants.

Week 2: Eigenvalues and eigenvectors.

Week
3: Solving linear systems. Rank
nullity theorem.

Week 4:
Limits and continuity in R^n. Partial and directional derivatives.

Week 5: Total Derivatives. Chain rule.

Week 6:
Higher order partial derivatives. Extremum problems in several variables.

Week 7: Hessians. Lagrange multipliers.

Week 8: Line integrals.

Week 9: Multiple Integrals.

Week 10: Green's Theorem

Week 11: Change of variables in multiple integrals. Spherical and cylindrical coordinates.

Week 12: Surface integrals

Week 13:
Stokes and Gauss's theorems.

Week 14:
Preceding material will take more
than 13 weeks to cover; I am not sure exactly where it will expand,

Basis for Grading

Each midterm and the homework will be 20% of your grade,
the final will be 40%. If you do well on the final, this grade can be
substituted for one of your midterms.

**Homework**

Homework is an extremely important part of the course. Whilst
talking to
other people about it is not dis-allowed,
too often
this degenerates into one person solving the problem, and other people
copying
them (often justified to themselves by saying "I provide the ideas, X
does
the details" - but the details are the key. If you can't translate the
idea into a real proof, you don't understand the material well enough).
So I
shall introduce the following rules:

(a) You can only talk to
some-one else about a problem
if you have made a *genuine *effort to solve it
yourself.

(b) You must write up the solutions on your own. Suspiciously similar
write-ups
will receive 0 points.

**Class**

I do expect you to come to class every day, and to participate in class discussions. I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.

The Discussion Sections are very strongly recommended, but I
understand some
students will have unavoidable conflicts with them.

Class etiquette: don't be disruptive or discourteous. No beeping, ringing, crunching, rustling, leaving early or arriving late. No texting, sleeping, checking your phone.

**Texts **
*Transition to Higher
Mathematics: Structure and Proof
*by Bob Dumas and John McCarthy ( **Available
free here**)

* Calculus,
Volume I* by Tom Apostol,
(Wiley) Second Edition, 1967

*Calculus,
Volume II* by Tom Apostol,
(Wiley) Second Edition, 1969

*Multivariable Mathematics
by*
Theodore Shifrin (Wiley 2005)

Note on the texts: The two books by Apostol
are
very expensive. They are not required texts for the course, though they
will be
useful.

I recommend buying used copies if you can.

There will be a copy of both Volume I and II on two-hour reserve at the
Library.

The book by Shifrin (also not required, and also, unfortunately, expensive) is less dense than Apostol.

**Additional Reading
**

*Vector Calculus* by J. Marsden and A. Tromba