10 Best Calculus Textbooks 2017
Spivak, M. Calculus [ 3rd Ed. 1994]
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates. In addition to extending the concepts of differentiation including the inverse and implicit function theorems and Riemann integration including Fubini's theorem to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy—Green , Ostrogradsky—Gauss divergence theorem , and Kelvin—Stokes , in the language of differential forms on differentiable manifolds embedded in Euclidean space , and as corollaries of the generalized Stokes' theorem on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results: [a]. Stokes' Theorem for Manifolds-With-Boundary. The cover of Calculus on Manifolds features snippets of a July 2, letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem i. Calculus on Manifolds aims to present the topics of multivariable and vector calculus in the manner in which they are seen by a modern working mathematician, yet simply and selectively enough to be understood by undergraduate students whose previous coursework in mathematics comprises only one-variable calculus and introductory linear algebra. While Spivak's elementary treatment of modern mathematical tools is broadly successful—and this approach has made Calculus on Manifolds a standard introduction to the rigorous theory of multivariable calculus—the text is also well known for its laconic style, lack of motivating examples, and frequent omission of non-obvious steps and arguments.
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