Course content: This course is devoted to the
study of the derivative and Riemann integral
of functions of several variables, followed by a treatment
of differential forms and a proof of Stoke's theorem
for manifolds in Euclidean space.
Students should have at this point enough computational background
(calculus of one variable and multivariable calculus -at least
2 and 3 variables), and I expect most of you to have taken Math 510 or
a comparable first course on rigorous analysis that includes:
sequences and series, the topology of metric spaces, continuous functions,
compacity, connectedness, the derivative
and Stieltjes integral of a one-variable function, sequences of functions and
Last semester we covered most of chapters 1-7 in
Principles of Mathematical Analysis, Third Edition by Walter Rudin.
MacGraw Hill Inc.
I promised this semester we will finish a few topics: equicontinuity, and special functions, then we will start in earnest with the material corresponding to this course. We will quickly review some linear algebra concepts and the topology on Rn. We then define the derivative of a mapping from Rn into Rm, and the chain rule. We discuss in detail the inverse function theorem which gives conditions on a differentiable function from Rn into Rn to have a differentiable inverse, and the implicit function theorem, which provides the theoretical justification for the implicit differentiation studied in calculus. We define the Riemann integral of a real-valued function of several variables and their properties. We discuss some simple version of Fubini theorem that allows to reduce computation of n-dimensional integrals to computation of iterated lower dimensional integrals. We also discuss change of variables, which is more complicated than the substitution technique in one-variable calculus. At this point we are ready to introduced manifolds, which are generalizations of curves and surfaces in higher dimensions. We define volume, boundary and integral of a scalar function over a manifold. The most important topic in the course is the interplay between the differential and integral calculus, the main theorem is Stoke's theorem for k-manifolds, which is a generalization of the fundamental theorem of calculus in one dimension, and in two and three dimensions bears the names of Green, Gauss (or divergence), and Stokes theorems. To carry on this program, differential forms must be introduced, and to manipulate them, some multilinear algebra is required. Differential operator of k-forms are introduced, and they replace the grad, curl and div in two/three dimensions learned in an undergraduate vector analysis course.
Prerequisites:Math 510, or permission from the instructor.
Required Textbook(s): Principles of Mathematical Analysis
by Walter Rudin. MacGraw Hill Inc. Third Edition, and
Analysis on Manifolds by James R. Munkres.
Munkres' book has been used in the past and all my colleagues report this is the book of their preference. It is a rigorous book, where calculations are done in detail and the geometry is emphasized.
I also recommend Vector Calculus, linear Algebra, and Differential Forms by John H. Hubbard and Barbara Burke Hubbard, 4th Edition, Matrix Editions 2009, and Mathematical Analysis. An Introduction by Andrew Browder, Springer 1996.
Homework: Homework problems will be handed out bi-weekly, and they will be graded and returned to you promptly. Problems from past real Analysis Qualifying exams will be weaved into the homework, hopefully by the end of the course you will have built a folder with solutions to most of those problems for future reference.
Grades: Grades will be based on homeworks, one exam, and possibly projects to be presented in class during the semester.
Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.
Return to: Department of Mathematics and Statistics, University of New Mexico
Last updated: 18 January 2010