Abstract: We prove a detailed sums of squares formula for two variable polynomials with no zeros on the bidisk extending previous versions of such a formula due to Cole-Wermer and Geronimo-Woerdeman. The formula is related to the Christoffel-Darboux formula for orthogonal polynomials on the unit circle, but the extension to two variables involves issues of uniqueness in the formula and the study of ideals of two variable orthogonal polynomials with respect to a positive Borel measure on the torus which may have infinite mass. We present applications to two variable Fej´er-Riesz factorizations, analytic extension theorems for a class of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.

Lecture A: Monday June 9, 8-10am

Lecture B: Monday June 9, 10:30am-12:30pm

The final will be cumulative, with roughly half old stuff and half new stuff. You are allowed to use a scientific (non-graphing) calculator.

Here are the finals I gave last quarter:

finalexam-8am

finalexam-11am

Here are solutions to one of the exams:

sample solutions (see comment below for a correction)

Also, consult the course schedule for links to all of the different sections that we have covered.

Here is the youtube page with all of the videos I have made.

Here’s the change of variables formula for double integrals:

where is the region we want to integrate over (which might be complicated, and that’s why we are using a change of variables), is a region in the -plane which gets transformed into under the one-to-one change of variables , and finally 

is the Jacobian which is equal to

.

Here are some recommended problems:

14.8: 1-4, 11, 12, 13-16, 21, 23, 24, 27, 29, Challenge: 33

Here is an introductory video in two parts:

Part one:

Part two:

To compute a triple integral using spherical coordinates, use the formula:

(this is maybe easier to remember as .)

Here are some recommended problems:

14.7: 1-4, 9-12, 17-24, 25, 26, 31, 32, 33, 37, 39, 43, 44, 53, 57

and here is an introductory video in two parts and an example of computing a triple integral using spherical coordinates:

Part One:

Part Two (very short):

An Example:

Cylindrical coordinates are really just polar coordinates with tacked on.

Give these problems a try:

14.6: 1-4, 8, 9-12, 19, 20, 21-24, 32, 33, 34, 39, 40, 45, 46, 49

and here is an introductory video:

Here are some recommended problems:

14.5: 1, 2, 5, 6, 9, 10, 17, 18, 21, 22, 27, 28, 29, 30, 35, 36, 37, 38, 41,

and here is an introductory video and an example:

Example of setting up a triple integral in two ways:

,

in shorthand:

.

Here are some recommended problems:

14.4: 1-6, 9, 10, 21-24  

and here is an introductory video:

Here are some recommended problems:

14.3: 1-4, 7, 8, 11, 12, 15, 16, 17, 18, 25, 26, 27, 28, 45, 46

And, here is an introductory video:

Here are some recommended problems:

14.2: 1, 2, 5, 6, 7, 8, 13, 14, 17, 23, 24, 26, 27, 37, 39, 43, 44

Here is an introductory video:

Here are some recommended problems:

14.1: 1, 2, 5, 7-10, 11, 14, 15, 21, 22, 25, 26, 31, 32, 33, 37, 38, 43, 49, 59, 60

And, here is an introductory video:

Here is an introduction to iterated integrals which you may find helpful: