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This is different to (Theoretical) Multivariable Calculus Textbooks as I want a classical treatment of line and surface integrals without the notion of a differential form.

Prerequisites: Paths, line integrals, the interior of a curve, orientation, surfaces etc. must be rigorously defined. The theorems of Green, Divergence, Stokes must be rigorously proven in some (or all) of their generality. Finally computational aspects of the theory must be kept to a minimum

You can suggest any book you want but I would prefer a set of notes (pdf format) accessible for free on the internet. Thank you for your answers

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Why do you not want use of differential forms? – Jebruho Nov 4 '12 at 18:40
I think I will leave that for when I study integration over manifolds. – Nameless Nov 4 '12 at 18:55
You really can't prove these results without differential forms. Even if you don't explicitly mention them, they will be lurking in the background. Just learn differential forms! It's not at all hard to learn the amount you will need. – Oliver Nov 4 '12 at 19:21
Really? Everything I am asking lie in differential forms? – Nameless Nov 4 '12 at 19:32

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