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Course of Action beyond Calculus 
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#1
Dec407, 09:34 PM

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I completed a study of Michael Spivak's Calculus recently. If I want to broaden my mathematics knowledge to the point where I can take a rigorous differential geometry course at a nearby university (I think they are using a differential geometry book also by Spivak), what should I do over the next year?
What (preferably rigorous) texts should I study and do a few problems from? I heard I should probably know some advanced calculus, what texts would you recommend to me for that? 


#2
Dec407, 09:40 PM

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calculus by tom apostol



#3
Dec407, 10:30 PM

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PF Gold
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If you completed Spivak, I would guess you know enough already!
If you want to broaden your knowledge, do some linear algebra or calculus with several variables. No need to go through calculus all over again. 


#4
Dec407, 10:34 PM

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PF Gold
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Course of Action beyond Calculus



#5
Dec407, 10:42 PM

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Isn't Apostol mostly just onevariable? I am not so sure about Apostol's Calculus, but I think Spivak very rigorously covered a lot about onevariable calculus.



#6
Dec507, 12:11 AM

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you could try calculus on manifolds by spivak, it's a small book, and should be easy to read at this point



#7
Dec507, 08:25 AM

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Do I still need linear algebra?



#8
Dec507, 08:30 AM

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#9
Dec507, 08:30 AM

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PF Gold
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A little bit of set theory might be useful. It might be covered in linear algebra books as an aside however.



#10
Dec507, 08:31 AM

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#11
Dec507, 11:55 AM

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Why else is good to change subjects? It's good because it will also increase your mathematical maturity. 


#12
Dec507, 12:30 PM

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You have to do linear algebra before you can do multivariate calculus anyway.



#13
Dec507, 02:09 PM

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I guess I'm a little confused!
So do I need to study: 1. Linear Algebra, 2. Multivariable, 3. And then I would be ready for Spivak's Differential Geometry Someone earlier mentioned something in diff. eqs... do I need that too? What books do you recommend for whatever sequence of study you would pursue if you were in my place? 


#14
Dec507, 06:19 PM

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From JasonRox:
Certainly studying something like linear algebra can broaden the mind since it is different than Calculus  new skills, new ideas. homeomorphism said: 


#15
Dec507, 11:14 PM

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I actually have pretty good experience with onevar Calc (through AP classes and Michael Spivak's Calculus).
I am only familiar with multivariable application, but I don't know the deep aspects: such as how and why Fubini's theorem works and where Riemann fails and Lebesgue works, etc. if you see what I mean. So I want to study multivariable in a deep/rigorous way. What books would you recommend? 


#16
Dec507, 11:24 PM

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#17
Dec507, 11:27 PM

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I don't even have that thorough of knowledge of Calculus and I would consider myself ready for Differential Geometry. (Of course, I have many other subjects in my background. The idea is that I touched many areas and am familiar with different aspects of mathematics. As well as I gathered some mathematical maturity in the mean time.) http://www.amazon.com/ElementaryDif...6918481&sr=82 Try that. It would requires you to know basic knowledge of linear algebra and multivarible calculus. Honestly, you're really not that far from reading this. If you're going to wait for school to teach you linear algebra and multivariable calculus, then yeah you're far from it. 


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