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  1. I guess this may seem stupid, but how calculus and real analysis are different from and related to each other?

    I tend to think they are the same because all I know is that the objects of both are real-valued functions defined on $\mathbb{R}^n$, and their topics are continuity, differentiation and integration of such functions. Isn't it?

  2. But there is also $\lambda$-calculus, about which I honestly don't quite know. Does it belong to calculus? If not, why is it called *-calculus?
  3. I have heard at the undergraduate course level, some people mentioned the topics in linear algebra as calculus. Is that correct?

Thanks and regards!

azimut
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Tim
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    λ-calculus is something completely different: see http://en.wikipedia.org/wiki/Lambda_calculus – lhf Apr 12 '11 at 01:45
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    I would say "calculus is to analysis as arithmetic is to number theory", including real and complex analysis under that umbrella. – Alex Becker Apr 12 '11 at 01:47
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    The term "calculus" can be used generally to mean something like "manipulation". The subject in math that we call calculus today was previously more well known by a longer name "calculus of infinitesimals", so named because at the time of its development, it was thought of as exactly that, the science of manipulating infinitesimally small numbers. It's in this sense that $\lambda$-calculus is named: it deals with the manipulation of "lambdas". See: http://en.wikipedia.org/wiki/Calculus_%28disambiguation%29 – matt Apr 12 '11 at 01:49
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    I think "calculus" in general means "to calculate". So, with this in mind, calculus uses the results of analysis to calculate things. Analysis is all the theory behind calculus. – Matt Gregory Apr 12 '11 at 02:00
  • @matt: thanks! I have heard that at the undergraduate course level, some people refer to topics in linear algebra as calculus. Is that correct? – Tim Apr 12 '11 at 02:05
  • @Tim: I've never heard linear algebra being referred to as "calculus". They are definitely integrated, though. You often use matrices of partial derivatives in linear algebra and you do calculus on vectors in multivariable calculus. But I don't really know why algebra isn't considered a type of calculus since it's used to calculate things too. I don't know. – Matt Gregory Apr 12 '11 at 02:12
  • A related question. – J. M. ain't a mathematician Aug 10 '11 at 06:51
  • You can use some extremely basic 'real analysis' to justify the nilpotent property, see here. –  Feb 14 '16 at 22:34
  • From my experience, the word Calculus tends to indicate an abstraction of ideas which can be used as rules in their own right. In the case of the standard integral/differential Calculus what's "under the bonnet" is limits and real analysis. But in the abstract sense we don't need to use these First Principles. They are assumed. Of course they must first be proven in order to use the Calculus with confidence, although this is not how it developed historically. Also, there are all kinds of Calculus, e.g. geometric, bigeometric, harmonic, even Umbral, etc. – pshmath0 Jan 04 '19 at 22:40
  • @Tim Terence Tao wrote in his "Analysis I" book: real analysis is the theoretical foundation which underlies calculus, which is the collection of computational algorithms which one uses to manipulate functions. (firs sentence of the first chapter). – Alessandro Jacopson Oct 20 '22 at 10:31

8 Answers8

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  1. A first approximation is that real analysis is the rigorous version of calculus. You might think about the distinction as follows: engineers use calculus, but pure mathematicians use real analysis. The term "real analysis" also includes topics not of interest to engineers but of interest to pure mathematicians.

  2. As is mentioned in the comments, this refers to a different meaning of the word "calculus," which simply means "a method of calculation."

  3. This is imprecise. Linear algebra is essential to the study of multivariable calculus, but I wouldn't call it a calculus topic in and of itself. People who say this probably mean that it is a calculus-level topic.

Qiaochu Yuan
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    we had courses analysis 1 and analysis 2 but the books had titles like Calculus. However these books were total 2000 pages too complex that any textbook for calculus i seen seemed childish too simple . So now i understand ,that books are analysis books. – GorillaApe Mar 03 '12 at 02:15
  • @Parhs: What books? – Tim Mar 03 '12 at 02:28
  • They are written in greek. I was wrong , the total pages are 2800 (2 theory and some problems and examples and 2 other only problems.) It uses literature from apostol,ayoub,birkhoff,comtet,ciang and lots of other[80 total].But they are extreemly hard to read. Even the most difficult textbook for calculus is easy compared to them.And they are given at engineering school – GorillaApe Mar 03 '12 at 02:54
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In Eastern Europe (Poland, Russia) there is no difference between calculus and analysis (there is mathematical analysis of function of real/complex variable/s).

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)

However, still this distinction is unnecessary:

  • (the most important) issues of "calculus" and "analysis" are very often linked together so that distinction is impossible (e.g. consideration of concept of limit in calculus due to Cauchy or Heine is actually the same as in analysis)

  • it makes artificial ambiguity in perception of mathematical analysis

  • it isolates common sense approach obtained from elementary mathematics and disables straightforward transition from elementary mathematics to higher mathematics

  • issues of "calculus" and "analysis" treated together enables acquisition of deeper understanding of subject by making extension from methods gained from elementary mathematics.

Community
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ICL
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    Sorry, but you must be confusing Anglo-Saxon countries and Western Europe. In France, there is no such thing as "calculus", and I suspect it's true for most of continental Europe. – Jean-Claude Arbaut Aug 26 '14 at 15:19
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    No such thing as "calculus" in Italy too. – LtWorf Jan 31 '15 at 09:38
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    In Romania there is also no "calculus" and I think it is better this way, it is more clear and less confusing. – yoyo_fun Jul 24 '17 at 10:25
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    No such thing in Germany, too. – Raphael J.F. Berger Feb 14 '18 at 14:59
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    There is definitely a distinction between "calculus" and real analysis in the USA. – James Hurley Feb 21 '18 at 06:38
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    I had the impression (mistaken according to these comments??) that in Europe, "calculus" (derivatives, integrals, etc.) was taught in the final year of secondary school? What the US might call "13th grade" ... students aged 18-19? – GEdgar Jan 04 '19 at 22:31
  • @GEdgar Students in the U.S., if they graduate on time, turn 18 in their final year of school, the twelfth grade (for many kids actually their thirteenth or fourteenth year in school since). If a student completes the bare minimum, he takes precal in twelfth grade; this is analogous to what Canadians call functions (idk about Britons). Most students, however, do more than the bare minimum, and therefore many students do take calculus in the twelfth grade. More and more are even entering accelerated programmes before secondary school and take calculus in eleventh grade. – gen-ℤ ready to perish Mar 30 '20 at 05:36
  • Funnily enough I also always thought that in Russian there is no " calculus" but only "mathematical analysis". It is actually very different, lots of "calculi": "Математический анализ" (Math analysis) is indeed the umbrella term but most of topic/subject names use the word "исчисление" (calculus). "Дифференциальное исчисление" "diferential calculus" etc. Nothing in math is called just "calculus"/"исчисление" in Russian. though. – Vitaly Osipov Sep 19 '21 at 02:21
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This is a purely Anglo-Saxon distinction. In most countries, however, there is no distinction between "rigorous" analysis and "non-rigorous" calculus. There are just different levels of analysis courses, e.g. "real analysis for engineers".

The term "calculus" itself just means "method of calculation". Even simple arithmetic is a kind of "calculus". What people in Anglo-Saxon countries refer to as "calculus" is actually just a short version of "infinitesimal calculus", the original ideas and concepts introduced by Leibniz and Newton. Nonetheless, even the lowest-level "calculus" courses usually refer to concepts that were introduced much later after Newton and Leibniz, e.g. Riemann sums (Riemann lived about 200 years after these two).

adjan
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    Are you sure it means method of calculation ? I was under the impression that it was derived from the old word for pebbles ie very small – Quality Aug 28 '16 at 19:21
  • @Quality That is just the etymology of the term. "Calculus" as a modern, English word means "method of calculation", e.g. "lambda calculus". – adjan Aug 31 '16 at 05:36
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As I understand the terms, calculus is just differentiation and integration, whereas real analysis also includes such topics as the definition of a real number, infinite series, and continuity. But perhaps I am out of date.

TonyK
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My take on this: One would use the word 'calculus' when one is applying the mathematical tools - chain rule, integration- by-parts, etc - to solve problems in science, engineering, and so on; whereas one would use the word 'analysis' when one is developing/justifying the same tools - proving the chain rule, inventing integration-by-parts, etc. I.e, analysis is what the pure mathematicians do, calculus is the product of analysis which engineers use.

PMar
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Calculus is about integration and differentiation. In real analysis we talk about Measure theory and lebesgue integral, proving theorems etc .And that introduces Topology , Functional analysis , Complex analysis , PDE and ODE etc .

telmen telmen
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    Welcome to stackexchange. It's good that you want to help by answering questions. But this short answer to an eight year old question does not add anything at all to the good answers already here. Since it uses the terminology of real analysis it would not help the OP in any case. So if you really want to help, look for new unanswered questions where you can contribute. – Ethan Bolker Jul 05 '18 at 01:36
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The classes in the U.S. are definitely separated by name, but the content of the 4 "Calculus" courses I took for electrical engineering wasn't quite as distinct as some are portraying it. We spent a substantial amount of time on sequences/series, limits, convergence, Taylor series, differentiability and integrability, etc., and related theorems. It was quite a bit more than "how to compute"; however, it wasn't nearly as rigorous as the proof-based real analysis class my son described from Caltech.

I'm sure that the engineering disciplines are a big reason for the distinction. The distinction may have other historical roots, but engineering schools sustain it. In the U.S., engineering (especially electrical) has changed in the last 50 years because the world is changing; instead of Fortran, it's incredibly helpful to know C++, Python, AND MATLAB; Newer high-speed transistors has shifted emphasis to RF circuit design classes; Increases in computer frequencies is pushing signal/power integrity to the forefront because-- in industry-- it's a dominant issue that barely existed in the 1980s. One EE prof. I spoke with said that electromagnetics, which formerly included distinct classes in electrostatics & magnetics, was condensed into 1 course that is barely adequate. There is an ever-widening list of courses/requirements and many students can barely finish a degree in 4 years; Math is just one of several areas receiving pressure to shed "unnecessary" requirements. My point is that real analysis would be good for engineers, but it's competing with many other math/non-math topics.

One can argue whether engineering should be a 5 or 6 year degree; but unless someone forces that upon everyone, which university will be the first to step forward and inform students that they must now pay for 5-6 years of school? In my opinion, economics is the sustained pressure to streamline courses and focus on practical "calculus" topics.

BartMan
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The issue here is that Calculus has several different meanings, some of them overlapping in context.

In mathematics, calculus can mean:

  1. a calculation or method of calculation (archaic, but motivates the other usages)
  2. a formal system in which symbolic expressions are manipulated according to fixed rules (and/or the rules themselves)
    e.g propositional calculus, predicate calculus, lambda calculus
  3. infinitesimal calculus,
    i.e. differential calculus & integral calculus
  4. By extension of 3, differential and integral calculus considered as a single subject in mathematical education (in addition to other pre-requisites taught as part of such courses)
  5. By extension of 4, a synonym of analysis (or some vague subset thereof) as a subject and/or branch of mathematics, since introductory analysis courses are generally based on 4.

- https://mathoverflow.net/questions/36758/difference-between-a-calculus-and-an-algebra
- What is the difference between a calculus and an algebra?
- What exactly is calculus?
- What do Algebra and Calculus mean?
- When do you call something "a calculus" vs. "a logic"?
- terminology: what is meant if someone writes "calculus of .."?
- What does "calculus" mean?
- Why "calculus"?

brazofuerte
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