In American universities two distinct types of courses are often called “Advanced Calculus”: one, largely for engineers, emphasizes advanced computational techniques in calculus; the other, a more “theoretical” course, usually taken by majors in mathematics and physical sciences (and often called elementary analysis or intermediate analysis), concentrates on conceptual development and proofs. This ProblemText is a book of the latter type. It is not a place to look for post-calculus material on Fourier series, Laplace transforms, and the like. It is intended for students of mathematics and others who have completed (or nearly completed) a standard introductory calculus sequence and who wish to understand where all those rules and formulas come from.
Many advanced calculus texts contain more topics than this ProblemText. When students are encouraged to develop much of the subject matter for themselves, it is not possible to “cover” material at the same breathtaking pace that can be achieved by a truly determined lecturer. But, while no attempt has been made to make the book encyclopedic, I do think it nevertheless provides an integrated overview of Calculus and, for those who continue, a solid foundation for a first year graduate course in Real Analysis.
As the title of the present document, ProblemText in Advanced Calculus, is intended to suggest, it is as much an extended problem set as a textbook. The proofs of most of the major results are either exercises or problems. The distinction here is that solutions to exercises are written out in a separate chapter in the ProblemText while solutions to problems are not given. I hope that this arrangement will provide flexibility for instructors who wish to use it as a text. For those who prefer a (modified) Moore-style development, where students work out and present most of the material, there is a quite large collection of problems for them to hone their skills on. For instructors who prefer a lecture format, it should be easy to base a coherent series of lectures on the presentation of solutions to thoughtfully chosen problems.
I have tried to make the ProblemText (in a rather highly qualified sense discussed below) “self-contained”. In it we investigate how the edifice of calculus can be grounded in a carefully developed substrata of sets, logic, and numbers. Will it be a “complete” or “totally rigorous” development of the subject? Absolutely not. I am not aware of any serious enthusiasm among mathematicians I know for requiring rigorous courses in Mathematical Logic and Axiomatic Set Theory as prerequisites for a first introduction to analysis. In the use of the tools from set theory and formal logic there are many topics that because of their complexity and depth are cheated, or not even mentioned. (For example, though used often, the axiom of choice is mentioned only once.) Even everyday topics such as “arithmetic,” see appendix G, are not developed in any great detail.