Syllabus: MAT 403-01 Advanced Analysis II. Spring 2013.

This course continues the work of MAT 401, proving important results of calculus. Besides being important for theory, the ideas we use (\(\epsilon-\delta\) arguments etc.) are important in practical applications because they enable people to estimate and control the size of errors when one uses approximate data.

The fundamental theorem of calculus, the theorem on the existence and uniqueness of solutions for ordinary differential equations ("Picard's theorem"), the implicit function theorem, and the inverse function theorem are highlights of this course.

The existence and uniqueness theorem says that if \(f(x,y)\) is a reasonably nice function then the ordinary differential equation \[ \frac{dy}{dx}=f(x,y) \text{ with initial condition } y(x_0)=x_0 \] has a unique solution in a neighborhood of the point \((x_0,y_0)\). This is important in science because differential equations are central in scientific applications.

The implicit and inverse function theorems are mathematically rigorous statements of the intuitive idea that if you have \(k\) independent equations in \(n\) variables \[ \begin{aligned} f_1(x_1,x_2,\ldots,x_n) &= y_1 \\ f_2(x_1,x_2,\ldots,x_n) &= y_2 \\ \vdots \\ f_k(x_1,x_2,\ldots,x_n) &= y_k \end{aligned} \] where \(y_1,\ldots,y_k\) are fixed, then the \(k\) equations impose \(k\) independent conditions on the \(n\) variables so there should be \(n-k\) "degrees of freedom" in the solutions. In other words, one should be able to parametrize the set of solutions with \(n-k\) variables. Also, if \(y_1,y_2,\ldots,y_k\) are allowed to vary then the solutions should vary in a nice way (continuously, differentiably, ...) with the numbers \(y_1,y_2,\ldots,y_k\). This generalizes a standard result from linear algebra which says that the dimension of the space of solutions to a matrix equation \(Ax=y\) is equal to the number of columns in the matrix \(A\) minus the number of independent rows. The link between the calculus version and the linear algebra version is the fact that differentiable functions can be approximated with matrices over small regions. We will study that fact too.

Texts

  1. Basic Analysis: Introduction to Real Analysis by Jiří Lebl. Download for free or purchase an inexpensive paper copy at http://www.jirka.org
  2. Introduction to Real Analysis by William Trench. Download for free or purchase at http://ramanujan.math.trinity.edu/wtrench/misc/index.shtml

Supplementary reference for help with logic and proof

Book of Proof by Richard Hammack. Download for free or purchase an inexpensive paper copy at http://www.people.vcu.edu/~rhammack/BookOfProof/

Prerequisites

MAT 401 Advanced Analysis I or equivalent with grade C or better. A course in linear algebra would also be a plus although it's not essential.

Learning Outcomes

Please see the Math Department Syllabus http://www.csudh.edu/math/syllabi/MAT403DeptSyllabus.html

Computer skills

Not required.

Grading Policy

Grades are based on homework, two midterm exams, and a final exam:

Attendance Requirements

I rarely take attendance but do come to class and participate. Students who miss class frequently almost never do well.

Policy on Due Dates and Make-Up Work

I don't plan to accept late work.

Assignments, Exams, etc.

Please see the course calendar (see link above), and check back at least once a week for updates.

Academic Integrity

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how I can best help you. All disclosures of disabilities will be kept strictly confidential. Please note: no accommodation may be made until you register with the DSS in WH B250. For information call (310) 243-3660 or to use telecommunications Device for the Deaf, call (310) 243-2028.