Spivak Calculus on Manifolds vs Apostol Volume II Next year I'll be taking a course in multivariable calculus. I was planning to study ahead of time using Spivak's Calculus on Manifolds since I had greatly enjoyed learning single-variable calculus from Spivak's Calculus. However, the course itself is taught from Apostol Volume II. What are the greatest differences between the two textbooks? Which of the two would you recommend?
 A: Having owned and read both of these texts, I suggest that you should stick with the course selection.  Calculus on Manifolds is completely different from Apostol's Calculus Vol. II and you would not be served by reading the former at this time.
In fact, without a background in multivariable calculus, it is doubtful you would even get beyond the first chapter of Spivak's text.  The prerequisites that are lacking include topics such as:


*

*Line integrals over a vector field

*Volume and surface integrals

*Stokes' and Green's theorems

*Differentials in higher-dimensional spaces

*Multivariable generalizations of the chain rule

*Vector and matrix algebra


Apostol will teach these to you at a level that can be appreciated by a mathematics undergraduate.  Spivak's text assumes familiarity with these concepts and shows how they generalize on manifolds, and how the fundamental theorem of calculus is a special case of Stokes' theorem.  The theory of differential forms is introduced.  I would not regard it as an undergraduate level text--that is not to say an undergraduate cannot appreciate the content, but rather, its terseness and treatment by Spivak is not suitable for an undergraduate textbook.
