Peng T. Mathematical Analysis Vol II. 2024

Textbook in PDF format

This is the second volume of a textbook for a two-semester course in mathematical analysis. This second volume is about analysis of multi-variable functions. The topics covered include Euclidean spaces, convergence of sequences, open sets and closed sets, limits and continuity, uniform continuity, connectedness, compactness, intermediate value theorem, extreme value theorem, partial derivatives, differentiability, chain rule, mean value theorem, first and second order approximations, local extrema, inverse function theorem, implicit function theorem, constrained extrema problems and Lagrange multipliers, Riemann integrals of functions of several variables, Jordan measurable sets, iterated integrals, Fubini's theorem, change of variables theorem, Fourier series and its convergence, Fourier transforms.
Besides calculus, it is also assumed that students have taken introductory courses in discrete mathematics and linear algebra, which covers topics such as logic, sets, functions, vector spaces, inner products, and quadratic forms. Whenever needed, these concepts would be briefly revised