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I am deeply sorry if this thread or discussion topic does not belong to this forum, but I have no idea on where to post this issue of mine.

Essentially I have finished a Calculus 1 course, but kind of more "advanced", as we viewed single variable differential calculus, differential equations (from variables separable to first order linear), elementary vector calculus and multivariable differential calculus (partial derivatives, implicit differentiation and optimization). However, now that university is over for the winter break, I was planning on doing some independent study as I completely adore math (I am planning to major in math).

The question is: what should I study over the break? The topics that we are going to cover in our Calculus 2 course are integration (applications and all the like, such as arc length, curvature and solids of revolution), series and sequences (up to power series) and multiple integration. I have basic experience with integration (if finding basic antiderivatives, and substitution rules counts as experience). I am really interested in the course material and i would like to make some use of my free time to make things smoother next term (and have some fun in the process). I was thinking of studying abstract algebra (I have already covered linear algebra, even though I had bad professors) or real analysis, but I have realized that I need to grasp the material in Calculus 2 to pursue something more complicated.

Thank you very much for your insight, and once again, I am very sorry if this topic does not belong to this discussion forum.

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Spivak's Calculus is a really good book. It is a theoretical (proof heavy) book that covers all of single variable calculus. You can then move on to his "sequal" Calculus on Manifolds which covers multivariable calculus. In the second book the first few sections cover multivariable differentiation and integration. The second part goes into more interesting things like differential forms, manifolds and stokes theorem. Another book for multivariable calculus is Calculus on Manifolds by Munkres which is more elaborate and detailed than Spivak's book.

You might also be interested in reading an introductory book on differential geometry. You will learn about curves (arc length and curvature) and other interesting things.

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  • $\begingroup$ The bad thing is that at the moment i can't honestly afford any textbook for the time being (i'm really tight with money right now). In my hands, i've got Stewart's Multivariable and Single Variable Calculus textbooks, but i'm not sure if they will be enough. If they are, with what should i start with? (keep in mind i'll have only 20 days for this) $\endgroup$ – arcbloom Dec 14 '13 at 4:19
  • $\begingroup$ @fogvajarash That's ok. There are a lot of good resources on the internet also. For example math.harvard.edu/~shlomo. Try the Advanced Calculus book. Its really big and covers a lot of material in linear algebra and calculus. So pick and choose what to study and go at your own pace. Chapters 3, 6, 8 will be the most relevant. $\endgroup$ – Pratyush Sarkar Dec 14 '13 at 4:43
  • $\begingroup$ That's really nice material! But wouldn't jump starting into differential geometry be kind of a bit tough? I mean, on which topic i should really focus on these few days that can actually expand my "mathematical maturity"? Thank you very much for the tips, i really appreciate them. $\endgroup$ – arcbloom Dec 14 '13 at 5:46
  • $\begingroup$ @fogvajarash May be this better suites your needs: ramanujan.math.trinity.edu/wtrench/texts/… . $\endgroup$ – Pratyush Sarkar Dec 14 '13 at 16:52

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