When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere between computational calculus and analysis? It becomes even more confusing when there are books like Thomas' Calculus (the old 3rd or 4th editions) which prove all the results, and books like Apostol's calculus which also prove all the results; but Apostol is deemed harder than Thomas. So is there also "easy proof-based calculus" and "hard proof-based calculus"?
Finally my last question is which would you recommend studying first for someone who has done single variable computational calculus but no multivariable calculus at all? Should I study computational multivariable calculus first before moving onto proof-based calculus? And if so, should I start with "easy proof-based…" or "hard proof-based…" (as described above).
Note: I apologize if you haven't seen Thomas' calculus book but it was pretty much the only one I could make the comparison with. The only other one I know that I would describe as "easy proof-based calculus" is Silverman's Calculus with analytic geometry.
Edit: Seeing as some people didn't understand the question I will try to clarify. By "proof-based calculus" I mean "calculus with theory" in the sense of the following MIT course: http://ocw.mit.edu/courses/mathematics/18-014-calculus-with-theory-fall-2010/ Since MIT also teach analysis, it seems to me there must be a significant difference between "proof-based calculus" and analysis. I tried checking the syllabus for both courses but the difference still wasn't all too clear, maybe because I have only done (computational) single-variable calculus and therefore don't have much of a base to make a comparison.
Best Answer
Perhaps the best way to think about the difference is as one of emphasis.
"Proof-based Calculus" would have the goal of doing Calculus but would justify the methods of Calculus (integration and differentiation) by proving their validity.
"Analysis" would have the goal of developing the theory of Calculus (and more than just Calculus)* at least partly for its own sake.
Consider the example of the real numbers. "Proof-based Calculus" is concerned with the real numbers only because it will need them for doing Calculus. "Analysis", on the other hand, treats the set of real numbers as an object worthy of study without having to consider further applications.
*Analysis is also much broader in scope than Calculus.