Prerequisites for Spivak's Calculus on Manifolds Hello I am a sophomore and I would like to know if Michael's other book named 'Calculus' is enough preparation for the Manifolds, if not please do tell what else should I be reading? I am taking an introductory course to topology next semester as well.
Thanks!
 A: You also need some topology, one semester is probably enough, and you need linear algebra. The linear algebra needs to be oriented to the theory of vector spaces and linear maps without always studying things in terms of bases and matrices. Linear algebra courses in applied math, engineering, statistics, etc. will likely not be sufficient. I recommend Hoffman and Kunze.
Multivariate calculus (e.g. Stewart) is helpful but not necessary. It's often taught as Calculus 3 in a typical sequence. I recommend it because theoretical texts, like the Spivak you are referring to, often deemphasize computation to the point that you forget that being able to calculate integrals and volumes is a big part of why we study this at all. If you use Stewart, you should note that the chapters on path integrals are actually about manifolds and are much harder than the rest of the book.
I'll also note that none of my professors thought there was really any good book on manifolds, which is a topic somewhere between real analysis and differential geometry, and arguably algebraic geometry (in terms of mathematical maturity). You might cross-learn from Munkres, or try to find open courseware notes or lectures on the topic, e.g. MIT or Stanford.
