Functions of Several Variables

The purpose of this book is to give a systematic development of differential and integral calculus for functions of several variables. The traditional topics from advanced calculus are included: maxima and minima, chain rule, implicit function theorem, multiple integrals, divergence and Stokes’ theorems, and so on. However, the treatment differs in several important respects from the traditional one. Vector notation is used throughout, and the distinction is maintained between n-dimensional euclidean space and its dual. By introducing convex and concave functions a more thorough treatment of extrema is possible. The elements of the Lebesgue theory of integrals are given. In place of the traditional vector analysis, we first introduce exterior algebra and the calculus of exterior differential forms. The book is suitable for a college course at the advanced undergraduate level. By omitting certain chapters, a one semester course can be based on it.