Math 204, Spring 2016

Honors Mathematics II

Instructor                 John E. M^cCarthy
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.
Homework 2 due February 2.
Homework 3 due February 9.
Homework 4 due February 16.
Homework 5 due February 23.
Homework 6 due March 1.
Homework 7 due March 8.
Homework 8 due March 22.
Homework 9 due March 29.
Homework 10 due April 5.
Homework 11 due April 12.
Homework 12 due April 19.
Homework 13 due April 26.
Homework 14.

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