Calculus on Manifolds (book)

Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965, ISBN 0-8053-9021-9) by Michael Spivak is a short (146 pp.) text treating analysis in several variables in Euclidean spaces and on differentiable manifolds. The book develops the classical theorems of advanced calculus, those of Green, Gauss, and Stokes, in the language of differential forms and in the context of differentiable manifolds embedded in Euclidean space. Calculus on Manifolds aims to present the topics of vector analysis in the manner that they are seen by a working mathematician, yet simply and selectively enough to be understood by strong undergraduate students. Even so, the book is famous, and at times, criticized for its terseness in the introduction of the intricate formalisms leading to the proof of Stokes' theorem on chains and manifolds.

A more recent textbook which covers these topics at an undergraduate level is the text Analysis on Manifolds by James Munkres (366 pp.). Although Munkres's text presents a more detailed treatment of the subject matter, the author acknowledges the influence of Spivak's earlier text in its preface.

The cover of Calculus on Manifolds features a copy of the original disclosure of the classical Stokes' theorem as it was written in a July 2, 1850 letter by Lord Kelvin to Sir George Stokes.