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Could anyone explain me what's the difference in terms of arguments between analysis and calculus? I borrowed two books from Apostol to have a quick look (Calculus I and Mathematical Analysis) and the arguments looks pretty much the same, the first one is focused more on "excercises" while the second one is more focused on the theorems that the first one doesn't prove.

I was wondering specifically if there's something that analysis allow to do while calculus doesn't.

In a practical problem, what Analysis allows to see/understand that calculus doesn't and viceversa.

user8469759
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  • Basically Mathematical analysis and Calculus are synonyms; at most, we can say that Calculus is the "elementary" part of Math anlysis. – Mauro ALLEGRANZA May 17 '16 at 08:27
  • See this question: Are calculus and real analysis the same thing?. – Eff May 17 '16 at 08:28
  • @Eff, I can understand that Calculus is taught to engineers and physicist while analysis is usually done in maths, however I don't understand what kind of problems can be tackled using Calculus rather than analysis. – user8469759 May 17 '16 at 08:38
  • @MauroALLEGRANZA, by elementary do you mean strictly subset of analysis? if yes does this mean that analysis is more powerful than calculus? – user8469759 May 17 '16 at 08:38
  • @lukkio From the linked question:

    "In my opinion this distinction is typical for Western countries to make the following difference:

    calculus relies mainly on conducting "calculations" (algebraic transformations applied to function, derivation of theorems/concepts by methods of elementary mathematics, computations applied to specific problem) analysis relies mainly on conducting "analysis" of properties of functions (derivation of theorems, proving theorems)".

     – Inazuma May 17 '16 at 08:54
  • Apparently differential equations are part of analysis but not part of calculus. – Heimdall May 17 '16 at 09:01
  • Also I would probably compare 'calculus' with the 'what' and analysis with the 'why' and 'how'. For example, for an engineer, they might be able to say that bricks are good building materials, but it is even more powerful for them to explain why. Some people hate mathematics because they think it's all about memorising formulas - and when you learn high school calculus it mostly is. But in further level courses, you learn about why things work (like the product rule, etc.) and do formal proofs, which is much more "powerful" (as everything in mathematics relies on proofs). – Inazuma May 17 '16 at 09:03
  • You've picked a bit of an outrider in Apostol's calculus, (or at least the edition I have from 1975) that text was a lot more rigorous and analytic than typical calculus texts which were mainly about manipulation of integrals, differential equations, multivariable calculus, series, complex analysis, etc. Analysis deals with the foundations, calculus deals more with applications. – user247608 May 17 '16 at 09:09
  • Can you please point me out an example of book which treat calculus then? I would like to understand the difference. – user8469759 May 17 '16 at 09:17
  • I mean when I did analysis/calculus at uni we had a course which was basically both analysis and calculus probably. But I'm curious about the difference. – user8469759 May 17 '16 at 09:20
  • I think a common distinction between analysis and calculus is the amount of rigour involved. It is hard to completely separate them because of how integrated they are (obviously), but there are plenty of opinions out there on the internet. – Inazuma May 17 '16 at 09:27
  • I view analysis as that part of mathematics that grew from the limit concept. The embryonic subject in analysis would then be calculus. Other subjects in analysis came later: complex analysis, metric spaces, Fourier analysis, etc. and you'd get some quizzical looks if you called these merely calculus. – zhw. May 17 '16 at 14:30
  • Analysis is calculus with muscles. – Lee Mosher Oct 20 '22 at 13:53

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According to Terence Tao: "real analysis is the theoretical foundation which underlies calculus".

His book Analysis I starts with:

1.1 What is analysis?

This text is an honours-level undergraduate introduction to real analysis: the analysis of the real numbers, sequences and series of real numbers, and real-valued functions. This is related to, but is distinct from, complex analysis, which concerns the analysis of the complex numbers and complex functions, harmonic analysis, which concerns the analysis of harmonics (waves) such as sine waves, and how they synthesize other functions via the Fourier transform, functional analysis, which focuses much more heavily on functions (and how they form things like vector spaces), and so forth. Analysis is the rigourous study of such objects, with a focus on trying to pin down precisely and accurately the qualitative and quantitative behavior of these objects. Real analysis is the theoretical foundation which underlies calculus, which is the collection of computational algorithms which one uses to manipulate functions.1

1Tao, T. (2016). Introduction. In: Analysis I. Texts and Readings in Mathematics, vol 37. Springer, Singapore.

Alessandro Jacopson
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