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.

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.

[2][3] For example, in order to state and prove the generalized Stokes' theorem on chains, a profusion of unfamiliar concepts and constructions (e.g., tensor products, differential forms, tangent spaces, pullbacks, exterior derivatives, cube and chains) are introduced in quick succession within the span of 25 pages.

[4][5][6] A more recent textbook which also covers these topics at an undergraduate level is the text Analysis on Manifolds by James Munkres (366 pp.).

[7] At more than twice the length of Calculus on Manifolds, Munkres's work presents a more careful and detailed treatment of the subject matter at a leisurely pace.

[8] Spivak's five-volume textbook A Comprehensive Introduction to Differential Geometry states in its preface that Calculus on Manifolds serves as a prerequisite for a course based on this text.

In fact, several of the concepts introduced in Calculus on Manifolds reappear in the first volume of this classic work in more sophisticated settings.