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Synopsis
This work provides a systematic examination of derivatives and integrals of multivariable functions; it is suitable for a one-semester course in what is usually called advanced calculus of several variables. The approach taken here is similar to that of the author’s previous text, "Continuous Functions of Vector Variables": specifically, elementary results from single-variable calculus are extended to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about functions of multiple variables.
Topics covered include differentiability and its relation to partial derivatives; directional derivatives and the gradient; curves, surfaces, and vector fields; the inverse and implicit function theorems; integrability and properties of integrals; and the theorems of Fubini, Stokes, and Gauss. The development of the vector-field theorems ties together material from many of the chapters and emphasizes the physical applications of the theory. Prerequisites for the reader include background in theoretical linear algebra, one-variable calculus, and some acquaintance with continuous functions and the topology of the real line.
Written in a definition-theorem-proof format, "Derivatives and Integrals of Multivariable Functions" has a conversational style and is replete with historical comments, questions, and discussions about strategy, difficulties, and alternate paths. Ideal for upper-level undergraduates whose next course will be real analysis, this book is a rigorous introduction to multivariable calculus that will help students build a foundation for further explorations in analysis and differential geometry.