Lectures on Advanced Calculus

This book is based upon handouts I have prepared for the advanced calculus course in department of Mathematics at Fu-Jen University during 1993-2008.
For undergraduate students majoring in Mathematics, advanced calculus is without question one of the most important courses in their curriculum. It not only introduces the students into the realm of analysis but also it provides building blocks for further study in all fields. However, for many students it is one of the most difficult courses because it requires mathematical abstraction and the expression in rigid mathematical language. It is all the more difficult because students must rely on textbooks published in other countries which are not typically suitable for our students. With almost one decade of teaching experiences in department of Mathematics I have come to a conclusion that the dryness and arduousness caused by language barrier is another major contribution to the difficulty of learning advanced calculus. For years it has been my mission to write a textbook that allows students to study and master the material on their own.

This book has adopted a fast-paced but rigid writing style. It covers the rudiments of elementary calculus and applies results from linear algebra in some chapters. The proofs for most of the important theorems i.e. inverse function theorem, implicit function theorem, Fubini theorem, Change of variables for multiple integrals and Green, Stokes and Gauss theorems in vector analysis etc. are provided. The way these theorems are presented and proofed in most advanced calculus textbooks is often over-complicated and tough to digest. The beginners are often intimated and impeded from going further. This book aims to answer these problems with a simple
goal to present the topic in a comprehensive but easy to grasp style. My expectation is that readers will easily pick up the material through self-study. Some important theorems come with examples for further demonstration so that the readers can effectively master the theorems and have a strong foundation for future research.