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Michael Spivak (b. 1940) is an American differential geometer and expositor of mathematics. In addition to this current volume (1965), he is also well known for his introductory but rigorous textbook Calculus (1967, 4th ed.
A First Course In Linear Algebra - Robert A. A First Course in Partial Differential Equations with complex variables and transform methods - H. A Quick Introduction to Tensor Analysis - R. A Treatise On The Differential Calculus with numerous examples - Todd Hunter. Brief - Istory of.
2008) and five-volume magnum opus A Comprehensive Introduction to Differential Geometry (1979, 3rd ed. Within a brief 146 pages, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus gives a rigorous account of multivariable calculus.
The first three chapters examine functions in Euclidean space and the generalization of differential and integral calculus to functions f: R n → R m. The final two chapters develop the modern machinery of differential forms and the exterior calculus to state and prove a sweeping generalization of the theorems of vector calculus, the generalized Stokes' theorem for manifolds-with-boundary. The classical theorems of Cauchy-Green, Ostrogradsky-Gauss, and Kelvin-Stokes alluded to in the subtitle are restated and proved as immediate corollaries thereof. This volume is aimed at the student who has completed at least one year of one-variable calculus and a term of linear algebra and who has, in the author's words, a certain 'rapport with abstract mathematics.' By presenting the contents of a third-term calculus course as seen by a modern mathematician, Spivak introduces the student to some of the language and concepts of differential geometry, allowing this text to serve as a prelude to his grand treatise on the subject.
WARNING: Be skeptical! First the good parts: this is one of the few texts to introduce undergraduates to multivariable calculus in a rigorous way, giving them a taste of differential geometry along the way. Too often, textbooks will bury the subtleties that arise on passing from the theory of real-valued functions on the real line to vector-valued functions in Euclidean space. Definitions given by these books are usually classical and will thus lack the precision needed to develop the theory with full rigor, and students are told to accept plausibility arguments in lieu of proof, a la Schey's 'div, grad, curl and all that'. This short text, essentially a pamphlet, is one of the first undergraduate texts to address the absence of works that present the theory of calculus in several variables from a modern yet elementary perspective.
This is the first of several celebrated textbooks by Spivak, who authored the present work when he was only 25. The youthful exuberance is apparent! It also accounts for the overly ambitious coverage and unrealistic expectation of students' capacity to absorb the layers of abstract definitions in the last two chapters, given the ostensible audience of the book (undergraduates exposed to a 'respectable' first year calculus course and one term of linear algebra), Because of its terseness, lack of motivation, and frequent appearance of typos and errors, chapters 4 and 5 cause some degree of frustration to all but the most capable students.
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(Indeed, Calculus om Manifolds has earned a reputation as being one of the most difficult math textbooks for an undergraduate audience.) These key portions include very little in the way of intuitive explanations or illustrative examples to help students make sense of the web of unfamiliar and abstract definitions that are presented. Too often, theorems are presented immediately after an unfamiliar term is defined and are proved by formal manipulation of symbols without a hint of motivation. On top of this, typos (sometimes in the definitions!) provide further stumbling blocks for comprehension. A student determined to fully assimilate the beautiful mathematics contained in this little book should keep in mind its themes (., curves and surfaces are approximated by linear transformations and problems taking place on manifolds are translated into ones occurring in Euclidean space by 'change of variables') to help think about what's going on behind the formalisms.
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He or she should view the book as a jigsaw puzzle and try to piece together the intricate collection of definitions and results, in spite of the aforementioned challenges. Theorems should be taken as questionable assertions: is it actually true? Is it false because of a typo? A missing hypothesis? Is the theorem true, but the proof wrong? Nothing should be taken at face value. And if one really gets confused by something, there are several online errata to consult.
A good alternative: The other text for undergraduates 'Analysis on Manifolds' by Munkres is in many ways better, but has the opposite problem of occasionally spoonfeeding the student with details he or she could fill in. Nevertheless, on balance, Munkres's text is much better pedagogically and generally error free. Of course, Munkres had the benefit of being able to use 'Calculus on Manifolds' as a model, a benefit which he acknowledges. These texts below should serve as good references, but may be unsuitable for pedagogical reasons (too elementary/too advanced). Rudin's 'Principles of Mathematical Analysis' presents analysis on several variables in a mere 96 pages.
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While all the main results are there, the denseness of his presentation is even more extreme, and he manages to present the results in full rigor without many key pieces of machinery, like tangent spaces, tensoriality of forms, or pullbacks, hiding these structures within ad hoc definitions and almost mystical formulas that appear out of the blue. Students can go back to admire his exposition,.after. they've seen it done in a more conventional way elsewhere. Below this level, Hubbard and Hubbard's text 'Vector Calculus, Linear Algebra, and Differential Forms' is an engaging account of more or less the same area, presented in a semi-rigorous way, with the more difficult concepts and proofs relegated to an appendix or not discussed at all.
However, this is a great text for the STEM student who (to use the authors' own analogy) wants to peek under the hood without taking the car apart. In greater generality and with many additional layers of dense abstract definitions is the treatment by Loomis and Sternberg in their text 'Advanced Calculus', which astonishingly was taught to genius-level freshman at Harvard for some time. Students with a strong interest in differential geometry can use this text as a spring board into several important ideas in this branch of mathematics.