Rating: 
Summary: Accessible
Review: What this little book lacks in depth (cf. Rudin) it makes up
for in accessibility. Certainly not the only book to have on
your shelf -- but it is a nice companion to other texts, when
you want to figure out what everyone else is talking about.
Reading it is much like talking face to face with the author.One deficit is that the book comes off a bit like brief
lecture notes -- a bit lacking not only in rigor, but
sometimes even lacking in clarity of definitions.
Rating: 
Summary: The Mathematician's Calculus
Review: 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.
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