Gayatri Chakravorty Spivak and Calculus on Manifolds Just heard of this woman for the first time in a while today (she spoke on campus some time ago), and I did a double take. I doubt there are many people who've heard of both, but did anyone else think from her unusual Anglo last name that she might be married to the great mathematician Michael Spivak? She's not, though...she's one of these postcolonial gobbledygook generators. Quite unlike the real Spivak's work - ah, I remember many a lovely day during my undergraduate years, manipulating manifolds and toting up tensor products... For those vanishing few who care about such things, a representative comment from Amazon that captures the elegance of Calculus on Manifolds:

When you are in college, the standard calculus 1,2, (maybe 3) courses will teach you the material useful to engineers. If you want to become a mathematician (pure or applied), you must pretty much forget the material in these courses and start over. That's where you need Spivak's "Calculus on Manifolds". Spivak knows you learned calculus the wrong way and devotes the first three chapters in setting things right. Along the way he clears all the confusion arising from inconsistent notation between partial derivatives, total derivatives, Laplacians, and the like. Chapter four contains the main objective of the book: Stokes Theorem. I think Spivak does a great job in minimizing the pain students feel when faced with tensor algebra for the first time, by carefully developing only what is essential. By first developing the notions of vector fields and forms on Euclidean spaces rather than manifolds, he eases the assimilation of these concepts. There is a slight price to pay by not developing the notion of tangent spaces in terms of germs and derivations (the modern approach), but this is quite justified for the level of the book. The student who completes chapter four (including the exercises) is well-equipped to study differential geometry. Chapter five is a brief introduction to differential geometry, a teaser if you will, for the amazing ramifications of the tools developed in the book. As Spivak remarks in the introduction, the exercises are the most important part of the book. Spivak rewards the students in the exercises by leaving many interesting developments to them like the indefinite integral of a Gaussian and Cauchy's integral formula. This book is a gem for the student of mathematics.

You might not believe it, but I use this stuff regularly in some of the work I do. The chain of connection is: calc. on manifolds -> high dimensional formal manipulation -> pattern recognition -> genetics. Many are partial to his calculus book, but I actually prefer either Anton (for first timers) or Apostol I & II (for rigor). Spivak's univariate calculus textbook is in that vast middle ground...his manifolds book, however, is in a class by itself. I never managed to get through his 5 volume work on differential geometry (I had other priorities), but a fair case can be made for that work as well... Funny how the name of a mediocre and entirely replaceable postcolonial nitwit can evoke memories of the eternal beauty of higher mathematics! Namespace conflicts have unpredictable consequences... Update A kindred spirit comments on Spivak's calculus book:

This book will give you a deep, crystal clear understanding of Calculus. It is also the perfect introduction to analysis or to rigorous math. Every single person on the Earth should have read this book. One day, when we have vanquished poverty and cured every disease, all of our genetically engineered, supersmart kids will read this book in elementary school.

Yes...soon ;)