Learning Smooth Manifolds This may come off being a silly question but I was curious if it would be possible for me to learn smooth manifolds from Lee's Introduction to Smooth Manifolds and learn multi-variable analysis along-side it. My current knowledge consists of real analysis (single variable), topology, abstract algebra, linear algebra. 
From other posts I've read that it is recommended to understand multi-variable analysis before understanding manifolds. But I want to know if I can just learn the necessary as I move through lee's book. 
 A: Putting whether or not you can aside. You really should learn multivariable analysis first. The logical order is


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*Linear functions on linear spaces ((finite dimensional) Linear Algebra)

*Non-linear functions on linear spaces which can be approximated by linear functions (Multivariable Calculus)

*Non-linear functions on non-linear spaces which can be approximated by linear functions and linear spaces (Smooth Manifold Theory)
Now you certainly can learn the third one without the other two, since it is a generalization. But, and this is important, Lee's book assumes that you've already done steps 1 and 2 and are comfortable with that material. So do yourself a favor and instead of going back and forth between multivariable calculus and Lee's book, do multivariable first and then go through Lee's book. You aren't saving yourself any time by trying to interleave the learning.
Of course, nobody will stop you from picking up Lee's book and trying to work through it. We are just saying you will understand things much better if you have a solid understanding of multivariable calculus. For example, you should know:


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*the inverse function theorem

*how derivatives are linear maps and how second derivatives are bilinear maps

*the topology of $\mathbf{R}^n$

*how to actually work with multivariable functions (this is huge because a lot of examples rely on this knowledge)

On the other hand, there are textbooks that develop the theory of mutivariable analysis along side of manifolds. Two examples are Spivak's "Calculus on Manifolds" and Munkres's "Analysis on Manifolds". The key differences are: these are aimed at the undergraduate level and they don't assume prior knowledge of multivariable calculus.
