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I study Computer Science, but our mathematic coures are a little bit to basic.

I'm looking for a "advanced Calculus" book for self-study that has a lot of exercises. The book should focus on explaining and not just containing a bunch of formulas.

what I already have learnt:

  • Matrices
  • Matrix transformation (e.g. Scale, Rotation, Translation and so on)
  • Vector calculation in $\mathfrak{R}^2$ / $\mathfrak{R}^3$ (Cross Product, Dot Product, Projection and so on)
  • Homogeneous coordinates
  • plane / sphere / line equation
  • derivation with one variable
  • integration
  • number series
  • limit
  • Complex Numbers / Quaternion (only rotation)
  • basic in discrete Mathematics

what I especially want to learn:

  • vector analysis
  • multivariable integration/derivation
  • Integral equation
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I am rather interested in acquiring this book

http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf

and the 2nd volume

http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich2.pdf

These two books are quite thorough, and rigorous, and go all the way from the (modern) foundations of calculus in $\mathbb{R}$, to discussing differential forms and their integration over suitable manifolds (a significant generalization of even multivariable calculus).

Edit: Links appear to be dead now...

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    $\begingroup$ The links do not work. $\endgroup$ – Anton Petrunin Dec 7 '16 at 16:32
  • $\begingroup$ Those links used to work, but I guess they no longer do... $\endgroup$ – Justin Benfield Dec 14 '16 at 7:33
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    $\begingroup$ It was a violation of copyright, no wonder they disappeared. But I recommend those books too. For more information: volume 1 and volume 2. $\endgroup$ – Jean-Claude Arbaut May 28 '18 at 13:09
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This book is online, and may be of use to you. It was apparently used at Harvard, and the material may be up to your standards of sophistication. http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf

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  • $\begingroup$ Thanks for the link. Nice book (and even nicer that it's on one of the authors' sites, so presumably perfectly legal rather than a pirated copy). Might not be exactly what the op's looking for -- sounds to me like he maybe wants a book that would be used in the engineering school rather than the math department. $\endgroup$ – John Forkosh Jul 3 '16 at 22:30
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Let me recommend Corral's Vector Calculus. The book is short and good, altho I would not call it "advanced". In addition it is written under free license (GFDL).

I used it to teach Vector Calculus at Penn State and made some minor changes which available here.

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