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

• Instructor: George Jennings
• Office: NSM A122
• Phone: (310)243-3592
• Email: gjennings@csudh.edu
• Website: http://www.csudh.edu/math/gjennings
• Office hours are posted on my website.
• Assignments and test dates will be posted on the Course Calendar http://www.csudh.edu/math/gjennings/403sp13/calendar403sp13.html . Please check frequently for updates.

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.

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.

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

• 40% of grade: Midterm Exams (20% each)
• 30% of grade: 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.