# 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 (1940 − ) is a short (146 pp.) text treating real analysis of several variables in Euclidean spaces and on differentiable manifolds. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions on Euclidean spaces, the book develops the classical theorems of advanced calculus, including those of Green, Ostrogradsky-Gauss (divergence theorem), and Kelvin-Stokes, in the language of differential forms and in the context of differentiable manifolds embedded in Euclidean space. The book culminates with the statement and proof of the generalized Stokes' theorem on manifolds-with-boundary:[1]

Stokes' Theorem for Manifolds-With-Boundary. If ${\displaystyle M}$ is a compact oriented ${\displaystyle k}$-dimensional manifold-with-boundary, ${\displaystyle \partial M}$ is the boundary given the induced orientation, and ${\displaystyle \omega }$ is a (${\displaystyle k-1}$)-form on ${\displaystyle M}$, then ${\displaystyle \int _{M}d\omega =\int _{\partial M}\omega }$.

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 exposed only to introductory courses in linear algebra and single-variable calculus. Even so, the book is famous, and at times, criticized for its terseness, in its treatment of the intricate formalisms of tensors and tangent spaces necessary for the statement and proof of the generalized Stokes' theorem on chains and manifolds.[2] Moreover, careful readers have noted a number of nontrivial oversights, typos, and misprints in the text.[3][4][5]

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

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.

## Notes

1. ^ The machinery of differential forms and the exterior calculus were developed by Élie Cartan, who discovered the general formulation of Stokes' theorem and published it in modern form in 1945: see: "The History of Stokes' Theorem" by Victor J. Katz (1979) Mathematics Magazine 52 (3): 146-156 (doi: 10.2307/2690275).
2. ^ Munkres, 1968
3. ^
4. ^ Axolotl, Petra. "Calculus on Manifolds Errata".
5. ^
6. ^ Munkres, 1991

## References

• Spivak, Michael (1965). Calculus on Manifolds (PDF). New York: W. A. Benjamin, Inc. (reprinted by Addison-Wesley (Reading, Mass.) and Westview Press (Boulder, Colo.)). ISBN 9780805390216. [A brief but rigorous and modern treatment of multivariable and vector calculus]
• Auslander, Louis (1967). "Review of Calculus on manifolds—a modern approach to classical theorems of advanced calculus". Quarterly of Applied Mathematics. 24 (4): 388–389.
• Munkres, James (1968). "Review of Calculus on Manifolds". The American Mathematical Monthly. 75 (5): 567–568. doi:10.2307/2314769. JSTOR 2314769.
• Botts, Truman (1966). "Reviewed Work: Calculus on Manifolds by Michael Spivak". Science. 153 (3732): 164–165. doi:10.1126/science.153.3732.164-a.
• Munkres, James (1991). Analysis on Manifolds (PDF). Redwood City, Calif.: Addison-Wesley (reprinted by Westview Press (Boulder, Colo.)). ISBN 9780201315967. [A somewhat less advanced but more careful treatment of the same topics]
• Loomis, Lynn Harold; Sternberg, Shlomo (1990) [1968]. Advanced Calculus (PDF) (Revised ed.). Reading, Mass.: Addison-Wesley (revised edition by Jones and Bartlett (Boston); reprinted by World Scientific (Hackensack, N.J.)). pp. 305–473. ISBN 0867201223. [A more sophisticated and detailed treatment in a more general setting]
• Hubbard, John H.; Hubbard, Barbara Burke (2009) [1998]. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (4th ed.). Upper Saddle River, NJ: Prentice Hall (4th edition by Matrix Editions (Ithaca, NY)). ISBN 9780971576650. [An elementary approach to differential forms with an emphasis on concrete examples]
• Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (PDF) (3rd ed.). New York: McGraw Hill. pp. 204–299. ISBN 0-07-054235-X. [A rigorous though unorthodox approach that sidesteps the typical abstract constructions]