Math 122 Multivariable Calculus - Fall 2007

Time and location:

The class meets on MTW-F 9:00- 9:50 am in Keyes 102

Office hours, calculus after hours and study groups:

Offic hours are Mon 11:00-11:50, Wed 11:00-11:50 in my office in MUDD 409

Calculus after hours with George Welch: Sunday through Thursday, 7:30 to at least 9:30 pm in MUDD 405. Starts Sunday 9/9/2007.

What to do if you want my help outside class and office hours: organizing a study group

Textbook:

you are recommended but not required to buy a book for this class.

1) Most of the time, our class will be following the exposition in Calculus, Multivariable by McCallum, Hughes-Hallet, Gleason, et al.

2) Another great book is Calculus on manifolds by Michael Spivak

Both of these books will (soon) be on reserve in the library.

3) Wikipedia also has a lot of entries about calculus.

Homework:

Homework is due on Fridays. There will be an online part and a written part for the homework. The written assignements will be completed as group projects in groups of three.

Exams:

There will be one midterm and one final exam.

Here is the practice midterm.

Here is the actual midterm with solutions and a key for how many points we gave for which part of the answer: page 1 page 2 page 3 page 4 page 5 page 6 page 7 page 8

Here is the practice final.

Grades:

Homework is worth 60% of the final grade, the exams are 20% each. Note that some of the grade for the written homework will be for style and exposition. You should be prepared to find yourselves writing sentences (rather than formulas) more than you are probably used to from your high-school math classes.

Schedule

Date	Summary	Assignments	Suggested reading
09/05	Administrative issues. The surface area of a circle: first questions (to be continued).
09/07	The surface area of a circle: more questions.
09/10	Paths in the plane, and how we could approximate their length. (Tangent vectors, velocity, displacement vectors).		Chapters 13.1 and 13.2
09/11	Paths in the plane: vectors, velocity, direction and derivatives. Finding the formula for computing the (arc)length of a path.
09/12	Arclength: review and group work on examples (to be continued).	First group assignement
09/15-16	Continuation of group work.
09/17-19	Functions in several variables, contour lines ("isoclines") and graphs. Examples. Practice to switch between the two points of view!		Chapter 12 in the book.
09/21	More examples. Among others, you have now seen paraboloids, cones, hemispheres, linear functions in several variables, and one- and two-sheeted (half-) hyperboloids.
09/24-26	Partial derivatives. You practiced in groups computing some of them.	First real online problem set (worth 100 homework points)	Chapter 14.2 in the book.
09/28-10/02	Directional derivatives and gradients. To understand their relationship to each other and the meaning of the gradient, you will need to learn about "dot products" and angles between vectors.	Project: find someone who you think is doing fascinating work (another professor, a TA, your best friend, your older brother, your mom ...), and ask them to tell you about one instance where they use mathematical techniques in their research/job. It need not be an application of calculus, and the person may be a mathematician, but they needn't be. Write a short essay about their answer. It will be worth 50 homework points. Due: Oct 12.	Chapters 13.3 and 14.4 in the book.
timeless	How to earn extra credit.	Spivak's book: many of the subjects that Spivak treats in his book (above) are the same that we are treating in class, just that he is discussing them a much more rigorous and logical framework. If you are curious about this, get together with a group of friends and start working through the book. Read it, try to do the exercises and some of the proofs that are left to the reader. Meet with me to discuss which parts you find easy and which parts you find difficult! See how far you get. It is a thin book, but you will find it quite a bit more densely written than your average textbook. You can earn up to 100 extra homework credits. (I will explain details in class.)	Spivak's book. (Library.)
10/03 - 10/05	Matrices and matrix multiplication. A matrix with n rows and m columns defines a linear function (through the origin) from m-dimensional to n-dimensional space. The Jacobian matrix of a differentiable function from R^m to R^n is the best linear approximation to this function. The chain rule says that the Jacobian matrix of a composite function is the product of the Jacobian matrices of the two functions.	Second online problem set (worth 100 homework points)	You can read the chapter on the chain rule, or look at the Wikipedia links (on the left) or both. Try to google the words, too.
Coming up	Second order derivatives and classifying critical values. The Hessian matrix.		The chapter on second order derivatives and the Wikipedia link on the left. Try to google the words, too.
Since the midterm.	Integration in several variables, polar coordinates, differential forms.	Problem Set (worth 100 homework points)	Polar coordinates are explained in the book, Chapter 16.4. Differential forms are explained in Spivak's book. You can also try Wikipedia, where you find a "gentle introduction" introducing the formalism that you should be able to handle (but without explaining properly what the symbols you are manipulating mean).

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