# Difference between calculus and analysis

It's somthing I always want to figure out, when did calculus start to be extended to analysis(I reformulate the question, the previous one"where one can draw a line to distinguish calculus and analysis, or there does not even exist such a line." was quite misleading).

As mentioned a lot in comments, analysis is a much border field than calculus, but the root could be traced back to the calculus in 19th century.

Besides, indeed infinitesimal calculus was proved in non-standard analysis, but it was invented until 1960s I think. And I don't know if it can replace all arguments in the theories developed after $\varepsilon-\delta$-definition and before the invention of non-standard analysis.

I will explain what I understand, please point out my mistakes.

The early stage (Newton and Leibniz)

They used infinitesimal, say $\mathrm{d}(\cdot)$ to describe change such as $\mathrm{d}x$ and $\mathrm{d}y$. And use $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y(x+\mathrm{d}x)-y(x)}{\mathrm{d}x}$$ to compute derivatives.

And let $y'$ be a shorthand notation for $\frac{\mathrm{d}y}{\mathrm{d}x}$, they defined integral as sum over infinitesimals $$\int y' \mathrm{d}x.$$ (I do not know how Newton and Leibniz defined integral. Maybe as $\approx\sum y(x_i)\Delta x$?)

19th century

People started worrying about the precision of infinitesimals. And the ratio of infinitesimals was replaced by limit (the '$\varepsilon-\delta$' definition).

In shorhand notation $$\frac{\mathrm{d}y}{\mathrm{d}x}:=\lim_{t\rightarrow 0}\frac{y(x+t)-y(x)}{t}.$$ 

While the notation was inherited, it did no longer hold the original meaning.

And Riemann established his formalization of integration.

Based on these, people started to work on functions defined in real number system (real analysis). And in the meantime, the properties of real number were intensively explored (set theory, continuum, etc.).

Later the concept of limit was further extended to more general spaces, such as metric spaces(generalized distance), normed spaces(generalized length).

So many branches of analysis such as measure theory(it's a part of real analysis. I put it here simply because I feel it is so important.), functional analysis, differential equations emerged.

Hence, roughly speaking, changing from infinitesimal approach to limit approach can be considered as the line separating calculus and analysis.

Interestingly, in modern calculus textbooks, they in fact loosely use analysis approach while they remain name themself as Calculus. Is this because they do not discuss real number system, which is the very base for the rest. And they only loosely argue 'taking limit by $\Delta x\rightarrow 0$'? I really get confused here.

Can I state that calculus is a study on real-valued functions with $\mathbb{R}^d$-valued argument? So one can loosely conclude that $$\text{infinitesimal and integral calculus} \subsetneq \text{real analysis}\subsetneq \text{analysis}.$$

Update again:

The question is much clear now. If calculus is understood as art of calculation, there is no more confusions.

Thanks for all dedications on this topic!

Since most of answers pointed out the linchpin for the question, I hope it won't cause any misunderstanding if I do not accept any of them.

At the end, I hope this post will help others in future.

Cheers.

• Defining derivatives using infinitesimals, Riemann integrals using infinite sums etc. can be made somewhat meaningful and formally correct by the means of nonstandard analysis... which is still called analysis, so there you have it. ;) On the other hand, I think analysis has a much broader scope than just calculus, just as you have written yourself, but not the other way around. Calculus is a subfield of analysis, the way I understand it. – tomasz Jun 15 '12 at 18:01
• In my mind, the etymology of calculus is important, coming from the Latin and originally meaning a pebble used as a reckoning counter (as on an abacus), and ultimately has the same root as calculate. Therefore, perhaps a defining feature of calculus is the calculation (of derivatives and integrals). (Real) analysis, on the other hand, is much more interested in the structure of the real numbers itself, and the functions defined on this system. Results in this area may find application in the calculation of derivatives/integrals (or bounds thereof), but it is not a necessary feature. – user642796 Jun 15 '12 at 19:10
• In my own professional field (electrical engineering, not mathematics) a similar question "What is the difference between circuit analysis and system theory?" was answered as follows. System theory is circuit analysis without the examples: circuit analysis is system theory without the theory. There are some analogies to this question since system theory (read: analysis) also deals with matters that have no direct bearing on circuit analysis (read: calculus) while circuit analysis methods can be used and understood without much specific and direct knowledge of system theory. – Dilip Sarwate Jun 15 '12 at 20:09
• (Continued) Put another way, if Mathematica were a human being, it could probably pass a calculus exam. I doubt that it could handle an analysis exam. Large computer programs carry out circuit analyses that human beings could not manage very well, but are nowhere near as good on system theory topics. – Dilip Sarwate Jun 15 '12 at 20:14
• Personally, I think there is a fuzzy boundary between the two, but the decisive break is when you stop looking at individual functions and their integrals and derivatives, and start looking at function spaces with their own topologies and metrics. – Mark Bennet Jun 15 '12 at 21:15

Remember, that the term "calculus" really stands for a method of calculation, especially one of several highly systematic methods of treating problems by a special system of ... notations. Think of vector and tensor calculus, calculus of variations, lambda calculus etc. Binding this term to integrals and derivatives is probably rather a pedagogical, than scientifical tradition.

The way I see it, "calculus" is just shorthand for "differential calculus" and "integral calculus". So it's mostly "calculus" when you are concerned with finding derivatives and integrals, and solving (easy) differential equations, with emphasis on the mechanical part.

"Analysis" is a much broader term, that includes the concepts and proofs concerning calculus (continuity, differentiation, integration), but many others, including for example Measure Theory and Functional Analysis.

As (I believe) the words are commonly used, calculus is the art of calculating, and analysis is the art of analysis.

The focus of a calculus course is on computation. Infinitesimal* calculus deals with methods for computing limits, derivatives and integrals symbolically, or by numerical approximations complete with error analysis. Even some/most/all of the proofs can be seen as exercises in manipulating approximations towards a goal.

*: I include differential and integral calculus in this category. Although the methods don't include 'true' infinitesimals as in the hyperreal numbers, the focus of these subjects is still usually along the lines of manipulating things by breaking them down into infinitesimal parts, or recombining the infinitesimal parts to yield an object of interest.

The focus of a real analysis course, on the other hand, is more about analysis -- breaking the subject matter apart into useful ideas. Methods of topology and measure theory are developed and applied, the theory and properties of derivatives and integrals are developed, and the ideas and structures involved are generalized beyond the 'simple' case of multivariate real functions.

In line with @ArthurFischer's comment: IIRC, in French "un calculus" is a small calculation. I think that broadly speaking, calculus courses and texts are (by tradition if not by intrinsic definition) about the rules and procedures needed to do calculations with derivatives and integrals: hence "integral calculus" and "differential calculus" (not to mention "Ito calculus" and "Malliavin calculus" if you want to get really fancy).

I tend to agree with Martin that "analysis", broadly speaking, tends to refer more to the body of work that looks at the proofs of these results, and how they assemble into a coherent whole that can be derived from basic foundations.

• I would add to this, that as far as I know the use of "calculus" for the basic differentiation and integration classes is a north-american thing. At least in my Spanish-speaking background, the first differentiation and integration courses are called "Análisis Matemático" (I guess no translation needed). – Martin Argerami Jun 16 '12 at 1:29