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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.

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    $\begingroup$ This question seems too vague to me. If I said "the difference between red and crimson is that crimson is darker", and then asked "where does dark red lie?", what would an answer to that question even mean? $\endgroup$ – Zev Chonoles Jul 2 '15 at 18:29
  • $\begingroup$ Also, with regards to your question "is there also 'easy proof-based calculus' and 'hard proof-based calculus'?", the answer is certainly yes, any subject whatsoever can be presented in various degrees of difficulty for the reader. $\endgroup$ – Zev Chonoles Jul 2 '15 at 18:32
  • $\begingroup$ @ZevChonoles: Semantics; as well as asking where it lies I also asked for the differences which is a rather concrete question. Or have you not heard of the phrase "proof-based calculus" before? Your example on the other hand doesn't even make sense: What shade of red do you mean by "red"? $\endgroup$ – user45220 Jul 2 '15 at 18:49
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    $\begingroup$ I think your question's wording is a bit vague, but the essence of it is very clear. Your experience is typical of many students who study calculus and this is primarily the fault of teachers and book authors of calculus (and syllabus designers). Mathematics education should always be provided with rigor (and not necessarily with formalism, in my opinion formalism is mostly detrimental). I would suggest you learn calculus from good books like Hardy's "A Course of Pure Mathematics" and your distinction between calculus and analysis will vanish. $\endgroup$ – Paramanand Singh Jul 3 '15 at 3:51
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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.

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