483:
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
488:
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
27:
513:) 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.
600:. Using this language, Cartan stated the generalized Stokes' theorem in its modern form, publishing the simple, elegant formula shown here in 1945. For a detailed discussion of how Stokes' theorem developed historically. See
442:
533:, 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
856:(Revised ed.), Reading, Mass.: Addison-Wesley (revised edition by Jones and Bartlett (Boston); reprinted by World Scientific (Hackensack, N.J.)), pp. 305–567,
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is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.
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638:"Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus | Mathematical Association of America"
758:
Auslander, Louis (1967), "Review of
Calculus on manifolds—a modern approach to classical theorems of advanced calculus",
979:, New York: W. A. Benjamin, Inc. (reprinted by Addison-Wesley (Reading, Mass.) and Westview Press (Boulder, Colo.)),
510:
230:
201:) to functions of several variables, the book treats the classical theorems of vector calculus, including those of
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Calculus on
Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (Mathematics Monograph Series)
461:
210:
190:
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186:
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218:
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serves as a prerequisite for a course based on this text. In fact, several of the concepts introduced in
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802:(4th ed.), Upper Saddle River, N.J.: Prentice Hall (4th edition by Matrix Editions (Ithaca, N.Y.)),
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A more recent textbook which also covers these topics at an undergraduate level is the text
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Calculus on
Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
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reappear in the first volume of this classic work in more sophisticated settings.
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Botts, Truman (1966), "Reviewed Work: Calculus on
Manifolds by Michael Spivak",
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containing the first disclosure of the classical Stokes' theorem (i.e., the
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800:
Vector
Calculus, Linear Algebra, and Differential Forms: A Unified Approach
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modern mathematical tools is broadly successful—and this approach has made
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The formalisms of differential forms and the exterior calculus used in
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836:
708:"Error in the statement of Thm. 2-13 in Calculus on Manifolds"
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A Comprehensive
Introduction to Differential Geometry, Vol. 1
181:
in
Euclidean space. In addition to extending the concepts of
118:
437:{\displaystyle \int _{M}d\omega =\int _{\partial M}\omega }
489:
profusion of unfamiliar concepts and constructions (e.g.,
956:(3rd ed.), New York: McGraw Hill, pp. 204–299,
999:(3rd ed.), Houston, Tex.: Publish or Perish, Inc.,
674:"Spivak - Calculus on Manifolds -- Comments and Errata"
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529:(366 pp.). At more than twice the length of
242:Stokes' Theorem for Manifolds-With-Boundary.
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873:(1968), "Review of Calculus on Manifolds",
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819:(1979), "The History of Stokes' Theorem",
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1020:(2nd ed.), New York: Springer,
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292:-dimensional manifold-with-boundary,
953:Principles of Mathematical Analysis
829:Mathematical Association of America
636:GouvĂŞa, Fernando Q. (2007-06-15).
423:
300:
14:
933:, Princeton, N.J.: Van Nostrand,
875:The American Mathematical Monthly
760:Quarterly of Applied Mathematics
785:10.1126/science.153.3732.164-a
689:"Calculus on Manifolds Errata"
540:Spivak's five-volume textbook
475:aims to present the topics of
160:of vector-valued functions of
1:
1018:An Introduction to Manifolds
544:states in its preface that
1075:
706:koletenbert (2012-10-02).
309:{\displaystyle \partial M}
231:generalized Stokes theorem
191:implicit function theorems
993:Spivak, Michael (1999) ,
596:were first formulated by
24:
563:Differentiable manifolds
219:differentiable manifolds
179:differentiable manifolds
901:Munkres, James (1991),
331:{\displaystyle \omega }
235:manifolds-with-boundary
1059:1965 non-fiction books
796:Hubbard, Barbara Burke
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162:several real variables
20:Calculus on Manifolds
16:Book by Michael Spivak
1049:Mathematics textbooks
1044:Mathematical analysis
919:Nickerson, Helen K.;
903:Analysis on Manifolds
594:Calculus on Manifolds
550:Calculus on Manifolds
546:Calculus on Manifolds
535:Analysis on Manifolds
531:Calculus on Manifolds
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507:exterior derivatives
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215:differential forms
1027:978-1-4419-7399-3
1006:978-0-9140-9870-6
986:978-0-8053-9021-6
963:978-0-07-054235-8
940:978-0-486-48090-9
930:Advanced Calculus
912:978-0-201-31596-7
863:978-981-4583-93-0
853:Advanced Calculus
848:Sternberg, Shlomo
809:978-0-9715766-5-0
779:(3732): 164–165,
383:{\displaystyle M}
284:{\displaystyle k}
262:{\displaystyle M}
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598:Élie Cartan
454:Lord Kelvin
227:corollaries
197:(including
152:is a brief
145:Description
61:Mathematics
1038:Categories
752:References
647:2017-04-09
602:Katz (1979
137:(1965) by
1016:(2011) ,
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950:(1976) ,
850:(2014) ,
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612:Citations
575:Footnotes
503:pullbacks
468:Reception
432:ω
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420:∫
413:ω
401:∫
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326:ω
301:∂
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154:monograph
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556:See also
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456:to Sir
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907:ISBN
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804:ISBN
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339:is a
189:and
119:OCLC
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