Math 412

Topics. This course is the second part of a rigorous introduction to mathematical analysis that begins with Math 411. We will prove theorems concerning multiple integrals, sets of zero measure, uniform convergence of general sequences and series of functions and power series in particular. We will study the Lebesgue integral and its properties in depth, general orthogonal expansions in inner product spaces, and the Fourier transform in particular. We will prove theorems about convergence of Fourier series and derive properties of functions from properties of their Fourier coefficients. We will finish by studying Taylor's theorem in multidimensions, Jacobians, gradients, properties of extrema, the implicit and inverse function theorems, and multiple Riemann and Lebesgue integrals.
Time. Classes meet Tuesdays and Thursdays, 10:00 am to 11:30 am, in Cupples I, room 216.
Prerequisites. Math 411, or the permission of the instructor.
Text. The lectures will follow the second half of the book Mathematical Analysis by Tom Apostol, second edition, published by AddisonWesley, ISBN 0201002884 (1974).
Homework. You are encouraged to collaborate on homework, and to work additional exercises from the relevant problem sections, although the homework grade will be based only on the exercises listed below. Please return your solutions to the instructor by the end of class. Problem sets will be assigned as follows:



Tests.

Grading. The course percentage will be a weighted average of the Homework percentage (30%), Midterm Test percentage (30%), and Final Examination percentage (40%). Students scoring higher than the class average can expect a grade of B or better. Students taking the Cr/NCr or P/F options will need a grade of D or better to pass.
Office Hours. See the instructor in Cupples I, room 105a, on Tuesdays and Thursdays from 9:3010:00am or 11:3012:00m, that is, immediately before or after class, or make an appointment with the instructor.