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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.


Updates:

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.

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    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.2012-06-15
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    @tomasz I was just about to mention that nonstandard analysis is not concerned here.2012-06-15
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    I always think of analysis as a branch of mathematics, and of calculus as a term which defines (more or less precisely) a range of what is covered in basic mathematical courses in analysis. This may be utterly wrong, but that's my impression of how the terms apply best.2012-06-15
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    @tomasz Yes, you are absolutely right. However, all those could be considered as stemmed from calculus. At the early stage of analysis, it was 'invented' to formalize infinitesimal calculus. My question might be clearer if it's rephrased as what is the sign for the initiation of analysis.2012-06-15
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    I don't really get the point of this classification. Analysis is such a general word, that you can include many other domains in it, like ODE, EDP and so on. For me calculus deals with derivatives and integration in the classical sense (Riemann, Jordan), while Real Analysis contains the Lebesgue integratioin and all the weak differentiation stuff.2012-06-15
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    You say in your question that "changing from infinitesimal approach to limit approach can be considered as the line separating calculus and analysis", however (as you also point out) most Calculus texts do use the limit approach and (as tomasz pointed out above) the infinitesimal approach can actually be rigorously proved. To go further than this, the infinitesimal approach can be *used* to define the limit concept, and anything in the limit approach can also be proved in the infinitesimal approach. Therefore, I don't think the infinitesimal vs. limit approach is the distinction.2012-06-15
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    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.2012-06-15
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    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.2012-06-15
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    (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.2012-06-15
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    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.2012-06-15

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