Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus 5th Edition

With its instantly recognizable cover, Calculus on Manifolds (1965) is a classic and for cultural reasons, every serious math/theoretical physics grad student should have read this deceptively slim volume by his/her first year of grad school. As one of the earliest texts on this subject, it's a admirable and ambitious attempt to modernize multivariable calculus (and the author wrote it at the age of 25!), but for several reasons, the text is of limited pedagogical value.

The most serious problem with this text is the number of errors. While they do not necessarily disrupt the overall picture the author is trying to convey, a few statements in the book are wrong in a substantial way, and many others require additional hypotheses or slight modifications. The astute student should carefully consider each statement the author makes before assuming its validity, and if false, spend some time thinking about how to make the statement true. Reading this book with this degree of scrutiny is, of course, an exceedingly time-consuming challenge, but could be very rewarding....

Compounding the frequent occurrence of errors is the terseness of the text. Nonobvious implications and arguments in proofs are frequently glossed over by words like "obviously" or "clearly" or merely strung together by a series of equal signs.

Problems with this book in detail:

1) a) The notation is poorly chosen, especially with respect to (alternating) forms and tangent vectors and spaces, such as the use of \Lambda^k(V) instead of the more standard A^k(V) or \Lambda^k(V^*) for the space of alternating tensors on V. In the addendum, he corrects this usage but claims that \Omega^k(V) has become the standard notation for this space. However, \Omega^k(M) is actually more commonly used as a notation for the vector space of differential k-forms on a manifold M.

b) Sometimes, even minor typos will result in considerable confusion on the part of the student. For instance, on p. 97, Spivak writes A\subset R^n, when he means A\subset R^m, leading to confusion about how to think about and visualize an n-cube. Another important typo occurs in Thm. 2-13 (p.43), a statement later used to prove Thm. 5-1. The reader can easily come up with counterexamples to this theorem as stated. To correct this theorem, f\circ h should be replaced by f\circ h^{-1}. Again, this is a small, but potentially bewildering error. There are numerous other smaller errors, like an unwarranted assumption in the proof of Thm. 5-2, for instance...

Combined with minor typos and/or poor choice of notation, the terseness of his writing makes some parts of the text very difficult to comprehend for the beginner. The definition of the pullback of a differential form, for instance (p. 89), is given in a particularly incomprehensible way because Spivak uses the same notation in two closely related (cf. p. 77) but inconsistent ways without explanation. Long story short, Spivak implicitly used the abbreviation f^* for (f_*)^* (where f_* is a map defined earlier on p. 89) without any comment to that effect. Although this is a conventional abbreviation in differential geometry, this notation is baffling for the beginner if not properly explained.

The errors/problems I mentioned here led to many hours of confusion. There are probably many others that I just glossed over without even recognizing any problem. (Indeed, there are several lists of errata online, as well as discussions on math stack exchange regarding these and other errors or sources of confusion.)

2) Spivak spends too long on the less interesting minutiae of derivatives and multiple integrals in chapters 2 and 3 (for example, his long discussion on the subtleties of Fubini's theorem), and still gets some of it wrong or imprecise (like the proof of the Riemann-Lebesgue criterion for Riemann integrability, for example).

3) In chapters 4 and 5, precious little is given in the way of intuition for the endless list of definitions relating to conceptually novel ideas like forms, tangent spaces, n-cubes, chains, and manifolds/diffeomorphisms. Perversely, he draws a star to define "star-like set" for the statement of Poincare lemma - a waste of space better spent on virtually anything else. The proof of Stokes' theorem is beautiful in its austerity, though utterly incomprehensible on first reading.

Despite the claim in the book's preface that only calculus and linear algebra are required as prerequisites, the truth is that mathematical maturity and sophistication, at least at the level of Chapters 1-8 of Rudin's "Principles of Mathematical Analysis", if not a bit more, are needed for the reader to really grasp the finer points of the text. A theory course in linear algebra, e.g. one based on Halmos' "Finite-Dimensional Vector Spaces" or Hoffman and Kunze's "Linear Algebra", is also needed. For instance, Spivak assumes familiarity of the reader with the concept of algebraic dual spaces. An understanding of the basics of point set topology would also help greatly.

The opaqueness of this book stem in part from the author's usage of an odd mixture of elementary and advanced terminology and techniques. Thus despite his efforts, to understand what Spivak has written with any depth requires some familiarity with the methods and heuristics of differential geometry. But of course, the whole point of the book's last two chapters was to introduce undergraduate students to some basic ideas from differential geometry! Thus, this book is a nightmare for self-directed learning -- a good professor or a friend who has done well in a modern differential geometry course (covering differential forms and manifolds) is essential. I suspect that a good number of the positive reviews for this book come from people merely browsed through or never even got to Chapters 4 or 5.

Munkres's "Analysis on Manifolds" treats similar topics in a slightly more concrete manner. The biggest difference is that he uses the submanifolds approach rather than (singular) cubes and chains which, at this level, are not particularly useful or essential anyway and only add to the confusion. It's more than twice as long as Spivak, which reflects a more accurate gauging of how much explanation is required for these difficult topics. It does occasionally err on the side of being too pedantic, his proofs tending to be long and wordy, and his exercises are on the easy side, but the work as a whole is orders of magnitude better than Spivak for teaching intuition. Highly recommended for the serious undergrad!

Lastly, it's worth mentioning that the contents of this little book serve as a teaser for the first volume of Spivak's five-volume Great American Differential Geometry Book "A Comprehensive Introduction to Differential Geometry". This volume gives a thorough, clear, and engaging exposition of the ideas presented in "Calculus on Manifolds", plus a lot more. However, the exposition is quite abstract and students might (at least initially) find the sophistication and the amount of material covered to be overwhelming. In this connection, Loring Tu's book "An Introduction to Manifolds" is a much more focused, less encyclopedic work geared towards beginning graduate students. It includes many well chosen examples to give some concreteness to the plethora of abstract concepts but does so without watering them down. John Lee's "Introduction to Smooth Manifolds" is more complete in its coverage, but is correspondingly less accessible but is still very lucidly written. Although Spivak suggests "Calculus on Manifolds" as a prerequisite for his subsequent tome, just about everything in the differential geometry portions of Calculus on Manifolds (chapters 4 and 5) reappears in it and is explained with greater clarity there.

In summary, "Calculus on Manifolds" is a book of historical interest and reading it is part of becoming immersed in the "culture" of mathematics. Furthermore, the ideas that appear in "Calculus on Manifolds" form the nucleus of the modern mathematician's conception of differentiable manifolds. Thus the serious student of geometry, topology, and/or analysis would do well to grok them! However, they are better explained in nearly every other textbook and one should not hesitate to use alternate sources to learn them. If you're an undergrad (I first encountered this book as a freshman), read it for fun, and if you find these ideas fascinating, try pursuing them seriously using Munkres or, if you're up for a challenge, any of the three graduate texts I mentioned earlier. If you're a grad student, revisit "Calculus on Manifolds" after finishing a real differential geometry course to appreciate Spivak's youthful efforts to make the subject accessible to mathematical neophytes.

I learned from this book as an undergrad; like the other reviewers mentioned, it had several errors. Some problems are difficult to decipher, while others are not practical due to the errors. Overall it's frustrating to learn from this book on your own. For self-study, Munkres is a better option. I personally referred to Munkres, Cartan, and Bachman.

The first half of the book covers a typical multivariable analysis course in a rather appreciable short amount of pages.

The second part of the book concludes with Stokes' Theorem and here is where the real frustration begins. For starters, weeks worth of exterior calculus is condensed into one slim chapter. You could try referring to other sources but the notation used by Spivak makes it frustrating to translate between authors. It takes a lot of patience until you finally are able to appreciate his development of the proof of Stokes' Theorem.

This book became famous because of the rather ingenious proof of Stoke's Theorem in Spivak's original course notes. The book was published, it seems, simply to make that proof and the associated machinery, which at the time were somewhat novel, widely accessible. That this is a book intended for differential geometers and topoligists considering teaching this materal, rather than a book intended for self study, is evinced by the staggering number of errors.

Many reviewers have claimed that the errors are predominantly typographical, and in any event do not affect the the development of the mathematics. This is untrue. The mathematical errors were so glaring that the book is now published with an addenda fixing some, but not all, of the incorrect mathematics. For instance, there is a circularity in the development of the integration apparatus which is not addressed in the addenda: in an attempt to bound the image of a set under a map, it is assumed that a diffeomorphism sends a set of measure zero to another set of measure zero; at this point in the text, nothing of the sort has been proved, and the text actually uses this special case in proving the general case that all diffeomorphisms send all sets of measure zero to other sets of measure zero.

Now, this is just one example. And anyone who has seen the material more carefully developed can spot this problem quickly, and prove the special case on their own. But certainly this makes the book unsuitable to the uninitiated.

I should say that Rudin's undergraduate text is no better in this regard. In Rudin, the change of variables theorem is proved in a very limited case. He then attempts to build the integration of differential forms over chains machinery as in Spivak, but in doing so he implicitly uses a much more general version of the change of variables theorem than was proved.

The third popular book at this level treating this subject is by Munkres. Many students complain that this book is pedantic, which may or may not be true. But it is the only one where Stoke's Theorem is rigorously and completely proved, without relying on circular or incomplete lemmas. To those students who claim Munkres is "too easy," and instead celebrate Spivak, I would simply point out that Spivak was obviously sufficiently obtuse to them that they missed the mathematical holes sprinkled throughout.

I scored the book 2/5 because the publisher ought to have requested the author release a second edition when it became clear the errors were sufficiently problematic that an errata-addendum was needed. It is still an unapologetically efficient resource for instructors interested in teaching this material.