''Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus'' (1965) by

Michael Spivak Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Per ...

is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.

Description

''Calculus on Manifolds'' is a brief monograph on the

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...

of vector-valued functions of several real variables (''f'' : R^''n''→R''^m'') and

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

s in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and

Riemann integration In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...

(including

Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the ...

) 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 form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

s on

s embedded in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, and as

corollaries In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...

of the

generalized Stokes theorem In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on ...

on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results: The cover of ''Calculus on Manifolds'' features snippets of a July 2, 1850 letter from

Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...

to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the

Kelvin–Stokes theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...

Reception

''Calculus on Manifolds'' aims to present the topics of multivariable and

vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

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. 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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

s, pullbacks,

exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...

s, cube and chains) are introduced in quick succession within the span of 25 pages. Moreover, careful readers have noted a number of nontrivial oversights throughout the text, including missing hypotheses in theorems, inaccurately stated theorems, and proofs that fail to handle all cases.

Other textbooks

A more recent textbook which also covers these topics at an undergraduate level is the text ''Analysis on Manifolds'' by

James Munkres James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including ''Topology'' (an undergraduate-level text), ''Analysis on Manifolds'', ''Elements of Alge ...

(366 pp.). 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. Nevertheless, Munkres acknowledges the influence of Spivak's earlier text in the preface of ''Analysis on Manifolds''. 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.

Footnotes

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References

* * * 'An elementary approach to differential forms with an emphasis on concrete examples and computations''* * 'A general treatment of differential forms, differentiable manifolds, and selected applications to mathematical physics for advanced undergraduates''* * 'An undergraduate treatment of multivariable and vector calculus with coverage similar to'' Calculus on Manifolds'', with mathematical ideas and proofs presented in greater detail''* 'A unified treatment of linear and multilinear algebra, multivariable calculus, differential forms, and introductory algebraic topology for advanced undergraduates''* 'An unorthodox though rigorous approach to differential forms that avoids many of the usual algebraic constructions''* 'A brief, rigorous, and modern treatment of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates''* 'A thorough account of differentiable manifolds at the graduate level; contains a more sophisticated reframing and extensions of Chapters 4 and 5 of'' Calculus on Manifolds*{{Citation, title=An Introduction to Manifolds, last=Tu, first=Loring W., publisher=Springer, year=2011, isbn=978-1-4419-7399-3, edition=2nd, location=New York, author-link=Loring W. Tu, orig-year=2008 'A standard treatment of the theory of smooth manifolds at the 1st year graduate level'' Mathematical analysis Mathematics textbooks Vector calculus 1965 non-fiction books