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At the present moment I find reading the chapters in Munkres' "Analysis on Manifolds" and Spivaks' "Calculus on Manifolds" to be a very tough read. By that I mean I often don't understand what the theorems and lemmas are truly trying to convey, and that then causes me problems when trying to solve the exercises. My plan is to build my mathematical maturity/strength by reading a good linear algebra book that focuses on proofs, then reading a multivariable calculus text(or notes) to get a basic understanding of the concepts like the "Inverse function theorem", and doing computations with them.

I did a quick review of the linear algebra concepts used in Munkres and Spivak and took a bit of time to get familiar with mulitvariable calculus topics--such as how the single variable definition of the derivative is modified to better suit the multivariable case-- and that hasn't made the process of engaging with munkres and spivak any easier. So i've decided to go through an entire linear algebra text making sure I understand every theorem and corresponding proof, and force myself to prove each excercise, then doing the same thing for a multivariable calculus text.

Before I begin this process, I want to know if doing this will make reading and solving problems in munkres/spivak a breeze. In other words, is my inability to properly engage with munkres/spivak due to my lack of proper engagement with a good linear algebra text, working out proofs in a single variable calculus text, and reviewing the multivariable calc text first? Or must I persist with studying munkres/spivak directly until it becomes clear?

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  • $\begingroup$ My suggestion is to pick up a textbook on real analysis (say, Rudin) and read it before reading Munkres/Spivak. $\endgroup$ – Moishe Kohan Dec 8 '17 at 15:34
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It's probably not a bad idea to have a good vector calculus book like Vector Calculus by Marsden https://www.amazon.com/Vector-Calculus-Jerrold-Marsden/dp/0716749920/ref=sr_1_2?ie=UTF8&qid=1512741057&sr=8-2&keywords=vector+calculus+marsden

or a good linear algebra book like Linear Algebra and It's Applications by Strang https://www.amazon.com/Linear-Algebra-Its-Applications-4th/dp/0030105676/ref=sr_1_2?s=books&ie=UTF8&qid=1512741101&sr=1-2&keywords=linear+algebra+strang

But I would recommend working through Multivariable Mathematics by Shifrin https://www.amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/dp/047152638X/ref=sr_1_1?s=books&ie=UTF8&qid=1512741159&sr=1-1&keywords=multivariable+mathematics+shifrin

This book contains a lot of the results you will find in Munkres and Spivak. I found it to be a great bridge between vector calculus/linear algebra and analysis on manifolds. The book contains lots of concrete examples and really gives you a feel for what is going on under the hood. A lot of the details that are usually swept under the rug are fleshed out in detail in this book. I really can't recommend it enough.

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  • $\begingroup$ Thanks for the detailed response! In what order should I read these texts? Once I finish Axler's "Linear Algebra Done Right", what should I start with right after? and Should I be reading these texts sequentially? $\endgroup$ – john fowles Dec 8 '17 at 14:21
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    $\begingroup$ Assuming you have not taken linear algebra or vector calculus yet I would recommend linear algebra->vector calculus->multivariable mathematics->real analysis->analysis on manifolds. Don't read Axler if it's your first time taking linear algebra. $\endgroup$ – Wintermute Dec 8 '17 at 16:13
  • $\begingroup$ and what would be a good real analysis text? $\endgroup$ – john fowles Dec 8 '17 at 22:22
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    $\begingroup$ The standard is Rudin's Mathematical Analysis. Almost everyone learns from this book at one point or another. I recently read Terrance Tao's Analysis I and II for a refresher on the subject. They we're some of the best math books I've read in a while. $\endgroup$ – Wintermute Dec 10 '17 at 15:28
  • $\begingroup$ Awesome! Thanks again $\endgroup$ – john fowles Dec 10 '17 at 17:45

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