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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!

7 Answers 7

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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.

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    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 school2012-03-03
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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.

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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).

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    @Quality That is just the etymology of the term. "Calculus" as a modern, English word means "method of calculation", e.g. "lambda calculus".2016-08-31
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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.

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    I had a course called "calculus" where we also were introduced to all most known sets of numbers, from the natural numbers to the complex ones, we talked about series and continuity and other things and in some instances we also had to deal with proving things, even though they were relatively easy, I guess. It could be that people exchange the terms sometimes... I think mathematics would be simpler if a committee responsible for standards would clarify this.2016-12-27
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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.

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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 .

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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.2018-07-05