Talk:Calculus

Enough already

Most of this page now consists of arguing whether calculus should be defined in terms of limits or not. This talk page is for a discussion about the article "Calculus", not the subject of calculus itself. Until edits are made, I suggest that, as per Arcfrk's suggestion of July 23 that we get back to making the article better rather than discussing this topic which, as far as Wikipedia policy would indicate, should not be done on this page. Whether calculus should or should not be defined in terms of limits is immaterial. The overwhelming majority of calculus texts are oriented that way, and until an alternative system gained popularity (that is, a similar number of published works have a different framework) we should not introduce fringe opinions.

I'm not going to contribute any more to a discussion of calculus itself: it's the article that matters. Make the edits; then we'll address those. Xantharius 23:06, 6 August 2007 (UTC)

Oh please, no one is going to make edits only to have these removed. This is your turf and you are the king. Long live Wiki sysops and administrators! 76.31.201.0 00:10, 7 August 2007 (UTC)

This isn't about sysops... this is about wikipedia policy. Most of the people you have been conversing with are not admins and your mention of them is not relevant the situation.--Cronholm¹⁴⁴ 00:22, 7 August 2007 (UTC)

Seconded, I plan on archiving this whole little debacle in two days. If anyone wishes to continue this conversation, either take it up on a talkpage or I can create a sandbox.--Cronholm¹⁴⁴ 23:37, 6 August 2007 (UTC)

An observation

the 'applications' section uses the word calculus too much. Am I crazy?

Quite possibly, the applications section was created in a piecemeal fashion so it doesn't have a coherent theme. I will try to streamline it.(remember to sign your posts!)—Cronholm¹⁴⁴ 11:02, 15 August 2007 (UTC)

Polynomial calculations

Some kind-hearted anonymous editor (not me) wants to add a computation of the derivative of xⁿ to the article. Rather than reverting back and forth, perhaps we could discuss it here?

I for one am opposed to adding the computation. I think it is a little to specific and concrete for an overview article like this. However, I am willing to listen to other views. 141.211.62.20 13:53, 15 August 2007 (UTC)

I agree. For this article, the computation of the derivative of $f(x)=x^{2}$ at a particular point is enough to demonstrate the technique. The article derivative might be a place for derivation of power rule, product rule, quotient rule, and chain rule. Rick Norwood 14:30, 15 August 2007 (UTC)

I think it should be included in the article since the power rule is a very fundamental computation for determining the derivative. Besides, the difference quotient is included, so why not the power rule?Dannery4 02:47, 19 August 2007 (UTC)

I think this article is already long enough. There's a "see also" link to calculus with polynomials, which covers the application of the chain rule that Dannery4 is interested in. Isn't that enough? DavidCBryant 10:47, 19 August 2007 (UTC)

Perhaps you all are right. I withdraw my opinion. As a side point though, I was commenting on the power rule, which is different than the chain rule.Dannery4 20:41, 19 August 2007 (UTC)

The difference quotient is included because it's more basic; it's the definition of a derivative. The computation for $f(x)=x^{n}$ is almost exactly the same as the one for $f(x)=x^{2}$ and hence it adds little to the article. Thus, it should not be included.

There was a small discussion on it a month ago; see Talk:Calculus/Archive 4#power rule for derivatives?. -- Jitse Niesen (talk) 13:57, 19 August 2007 (UTC)

The so-called "power rule" is just a special case of the product rule. (Did I say chain rule? Oops! I should have said "product rule".). In other words, once I know the product rule, then the "power rule" is easily derived by induction on the exponent (i.e., (x²)' = x + x = 2x, etc). I guess you can say it's "different", but to me the two rules look like more or less the same thing. DavidCBryant 22:30, 19 August 2007 (UTC)

The Calculus

Isn't calculus often called "The Calculus"? Zginder 23:43, 7 September 2007 (UTC)

Not as far as I know. —METS501 (talk) 03:03, 8 September 2007 (UTC)

Actually, you may be right. See for example this page which refers to it as "the calculus". —METS501 (talk) 03:04, 8 September 2007 (UTC)

This is an older usage, not common today. Rick Norwood 13:03, 8 September 2007 (UTC)

Maths professors sometimes still refer to it as the calculus. Zginder 13:36, 8 September 2007 (UTC)

The full name is "The calculus of infinitesimals" (cf "The calculus of variations"). I suppose one could drop "infinitesimals" and keep the definite article, but in itself, it does not make sense. I think it would be a good idea to discuss the etymology of the term, perhaps, giving some time scale of what was the subject called in different times. Arcfrk 22:01, 8 September 2007 (UTC)

This issue has already been discussed in archive 2 (here), but Arcfrk's suggestion would fix the problem for posterity (ie people asking the "The" question once in a blue moon). So I guess the question becomes... who would like to trace/(find someone who already has traced) Calculus' etymology from Newton and Leibniz to the present day? Then put it in an article (Calculus or History of calculus)... Any takers? —Cronholm¹⁴⁴ 08:27, 10 September 2007 (UTC)

Perhaps the choice of "Calculus or "The Calculus", in modern usage, is largely dictated by the speaker's or writer's nationality? I know in the US we tend toward informality in speech, which would lead to dropping the "The", I think. We also aren't big on Mr. and Mrs. these days, and workers often address their employers by their first names. I think in some other countries this is very different, with much greater emphasis being placed on formality and status-based rules of propriety, and greater respect being given to, well, greater respect. Many of us here are somewhat uncomfortable being addressed with a title and our last name, given the prominence in our culture of ideas of equality and populism, preferring a simple first name form of address, and perhaps we unconciously believe that branches of learning such as Calculus would experience similar discomfort if burdened with the distinction of a formal title! 68.46.96.38 10:19, 16 September 2007 (UTC)

Yes, I'm a latecomer to this discussion, but I did (believeitornot) come to this page looking for an explanation of "the calculus". I think a note in the main article would be most helpful. (I'd add it, but I am clearly not qualified. :) ) CSWarren 19:30, 12 October 2007 (UTC)

BCE and CE versus BC and AD

Someone changed all of the BCs and ADs to BCEs and CEs, respectively, and then they were reverted. The "common era" versions are, as far as I was aware, the less objectionable. Why were they changed, and is there a Wikipedia policy on this? It's not a religious article referencing Christianity: is there a good reason for having BC and AD? Xantharius 03:43, 14 September 2007 (UTC)

The policy accepts both but discourages reverts back and forth; this issue has come up before(look at the history and archives), and I don't care as long as we just stick with one. I believe that BC and AD were the original notation used on the article, so I would like to leave it at that to discourage rewarding wholesale changes of this variety. —Cronholm¹⁴⁴ 05:28, 14 September 2007 (UTC)

As I've mentioned before, there are people who go through Wiki changing BC to BCE and other people who go through Wiki changing BCE to BC. There is no way to stop them, so I think it is best to just ignore them. It keeps them off the streets. Rick Norwood 13:21, 14 September 2007 (UTC)

Can a clear definition be derived for Calculus?

This would be integral to improving the quality of this article. It should be of the form: Calculus is the branch (or field?) of mathematics which does such-and-such actions. The following is perhaps quite flawed, as I am not really knowledgeable about calculus, but perhaps it could be used as a starting point, and cleaned up?: "Calculus is the branch of mathematics which deals with the behavior of numerical values as they change." For a definition to be useful, it should be possible, using the definition, to differentiate if any specific thing is part of Calculus, or if it is not. If a clear definition of the term "Calculus" can't be achieved, can we really say "Calculus" is a meaningful term? 68.46.96.38 10:55, 16 September 2007 (UTC)

The problem is that "Calculus" is really a college course rather than a branch of mathematics. Many calculus textbooks say something like what you say, "Calculus is the mathematics of change," and I have no objection to using that definition in the article.

In fact, what is called "calculus" in undergraduate education is called "analysis" in graduate mathematics, and is one of the three major branches of abstract mathematics that follow foundations: topology, abstract algebra, and analysis. These three branches are, roughly, the study of point sets, the study of number sets, and the study of functions. "Calculus" is the undergraduate analysis course that studies functions but sweeps some of the more difficult problems under the rug.

I've done a minor rewrite of the intro to attempt to make this clear. Rick Norwood 14:03, 16 September 2007 (UTC)

In my opinion, calculus is clearly not a branch of mathematics, it is simply a bunch of rules for calculating integrals and derivatives (and some sequences). Looking up the word calculus on wiktionary gives "Any formal system in which symbolic expressions are manipulated according to fixed rules" (as the general meaning of the word) and as an example it lists "vector calculus", and it is really in this sense that "the" calculus should considered. I guess what I'm trying to say is that the first line should probably be something more like "In mathematics, calculus is a set of rules for computing derivatives, integrals and certain other limits, and constitutes..."; though this probably isn't great either. But I definitely think that it shouldn't be called a branch of mathematics. RobHar (talk) 10:01, 26 November 2007 (UTC)

A word means what the people who commonly use the word intend it to mean. The current meaning of calculus is that branch of mathematics which considers limits, derivatives, integrals, and infinite series. While some teachers only teach rules (this is called "cookbook calculus") most prove theorems, though not as many theorems as we once did. All major calculus books include theorems and proofs. Vector calculus is not about, for example, dot and cross product -- that's Linear Algebra -- but about integrals and derivatives of vector valued functions. Rick Norwood (talk) 14:25, 26 November 2007 (UTC)

What would you say is the difference between Calculus and Analysis?  --Lambiam 15:34, 26 November 2007 (UTC)

The branch of mathematics that considers limits, derivatives, etc. is analysis. The set of methods used to compute these things when considering a very specific class of functions of one real variable is what calculus is (one could include multivariable calculus and vector calculus). There was probably a point in time when calculus was a branch of mathematics, when people were proving new things and developing the methods, but it's probably been a while since there's been an article written on calculus or an NSF grant given out for calculus research. Similarly, there was a point when the theory of determinants could probably have been considered as a branch of mathematics, but math evolved and this became a specific set of results in the branch algebra called linear (or multilinear) algebra. Though it is true that some theorems are proved in calculus classes, most of the time this is handwaving, and calculus assignments basically never include questions about proving a theorem, and are definitely focused on seeing if the students are able to apply the rules taught to them. RobHar (talk) 20:11, 26 November 2007 (UTC)

You and Rick are basically saying the same thing. RobHar, I agree with what you are saying to a point, but calculus is a subset of analysis, just the introduction to it basically. And the amount of proofs involved in a calculus class varies widely from none to a fairly sophisticated level, depending on the institution and level (regular, honors, etc). I guess I have to agree that analysis would be the proper branch of mathematics and calling calculus that is stretching it a bit. I think it's possible to meet in the middle though. Perhaps, simply moving the sentence "In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions." to the end of the first sentence of the article would cover it. That way the reader sees that analysis is the broader concept right away. Add to that a way to make it clear analysis is the branch and calculus is a subset of/introduction to it and that should encompass what both of you are saying. - Taxman ^Talk 00:09, 27 November 2007 (UTC)

Hey Taxman, thanks for weighing in. I guess I just don't quite view it that way. For example, many (most?) places have calculus courses and analysis courses with pretty much the same course description (ie the subject matter covered overlaps, but the point of view is different), and if you look at places like Harvard or Princeton, many (most?) students that are going into math don't even take calculus and instead go straight to analysis, because knowing calculus isn't necessary to understanding analysis. I do view calculus as a subset of analysis, but I do not view it as a logical beginning (in terms of subject development, that is) of the subject, but rather a first example of concrete calculations. Like how modular arithmetic is a good introduction to rings and ideal, so one might begin an abstract algebra course by discussing Z/n and doing many calculations and theorems, but one could have simply started by defining rings and ideals. I guess the one thing I'm not sure about is that it seems calculus courses in Europe are more rigorous than ones in North America, perhaps we are having a disagreement that is based on regional differences. I guess, in my experience taking many calculus and analysis classes, and in teaching, the difference between regular and honours calculus classes has mainly been the difficulty of the problems, perhaps a bit the level of rigour, but proofs were never expected to have been understood for exams. RobHar (talk) 03:07, 27 November 2007 (UTC)

I don't know about many places. I can only tell you about the university I went to. The honors calc sequence was pretty theory heavy and proofs were very much demanded, on exams as well. Of course there was some cookbook too it, since it wasn't the purely theoretical class, which was available as well. The regular calc sequence was entirely cookbook. - Taxman ^Talk 14:56, 28 November 2007 (UTC)

Once again, it is not our place to write articles based on our low opinion of the American school system, but rather to report what standard sources say. I appreciate Taxman's effort to mediate, but limiting "calculus" to functions of one (real?) variable won't do. The course I'm teaching now covers functions of several variables and is called "calculus" and taught out of a book called "calculus", as was the course I took as a Freshman at M.I.T., where the textbook was Calculus and Analytic Geometry by Thomas.

Mathematics includes not only areas of current active research but also the discoveries of the past. Calculus does not stop being a branch of mathematics even if it is no longer an area of active research (which the non-standard analists would dispute). Rick Norwood (talk) 20:23, 27 November 2007 (UTC)

Just a note, sorry to insert here. I wasn't the one that limited calculus to one variable. I agree with you on that point Rick, I just didn't happen to specify my disagreement with that point of Rob's. - Taxman ^Talk 14:56, 28 November 2007 (UTC)

Hey Rick Norwood, I don't really think you're addressing my points. To address yours, firstly I do not have a low opinion of the American school system, and my opinion on that subject has nothing to do with what I'm trying to talk about (and I'm also not sure why the sentence begins with "once again"...) . Perhaps your next comment is only about Taxman, but I included above that multivariable calculus and vector calculus (and probably some complex integral techniques) can be considered calculus (and I do consider them calculus). I do believe that mathematics includes calculus (as I mentioned above that it was "a subset of analysis"), but I definitely disagree that calculus is a branch of mathematics. There was probably a point where calculating the area of a disc was a branch of mathematics, or the study of quadratic forms, or as I mentioned above the theory of determinants, but these branches became parts of bigger more general branches. Perhaps the non-standard analysts are studying how to formalize certain manipulations in calculus such as differentials, but this is not calculus, just like the study of the formal system of peano's axioms is not grade school arithmetic.

The points I'd like you to address are the following, and if they are addressed and I'm shown to be wrong, I'll have no problem agreeing with you.

1. You mentioned standard sources for the definition of calculus, which of the many included in the article say calculus is a branch of mathematics?

2. How do you account for the fact that universities (including MIT, your alma mater) have both courses called calculus and courses called analysis that cover the same material? This would lead me to believe that standard sources (namely university math departments) consider these to be different things.

3. Do you think that most calculus courses include a nontrivial amount of exam questions testing the student's understanding of proofs of the theorems, and/or their capability to produce more proofs?

4. Do you believe that the theory of determinants is (still) a branch of mathematics? You address this by saying calculus doesn't become not a branch of math, but you seems to say more generally no branch is ever demoted. Perhaps we're just disagreeing on the term branch.

My main actual disagreement with what Taxman said was a minor one, and involved the fact that a view calculus as a subset, not the beginning of the branch of analysis (in a logical sense, not a historical sense). What I'd like to see at the moment is something meaning "Calculus was an important branch of mathematics for hundreds of years and remains a part of mathematics as a subset of the modern branch of Analysis." (which is, I think, along the lines of what Taxman was suggesting). Thanks. RobHar (talk) 23:17, 27 November 2007 (UTC)

Would it help if we use "a field of mathematics", a formulation also used in Differential calculus, Calculus of variations, Vector calculus, and many other articles?  --Lambiam 08:22, 28 November 2007 (UTC)

I have no problem with "field" of mathematics or "area" of mathematics, but I prefer "branch" because it describes the role of increasing specialization in mathematics. The first "branch" is pure vs. applied, then pure branches into geometry, number theory, and analysis, then analysis branches into real and complex analysis, and so on. My own branch (twig?) is knots on the double torus.

So, why not title this article "analysis" and redirect "calculus"? Because calculus is the more common word. Articles like this are of value to laypersons, not to mathematicians. To a layperson, the first thing they need to know about any abstract knowledge is: into which major category does it fall (mathematics), what are the prerequisites for learning it (algebra, geometry, and analytic geometry), and what are its applications (advanced math, science, engineering, computer science).

Now, to RobHar's four questions. 1) The first reference I happened to pick, The Concise Columbia Encyclopedia, says, "Calculus, branch of MATHEMATICS that studies continuously changing quantities." 2) Calculus and analysis do not cover the "same" material. Analysis builds on calculus. From Royden, "Real Analysis", "It is assumed that the reader already has some acquaintance with the principal theorems on continuous functions..." 3) As I've said before, what calculus is does not depend on how well or badly it is taught. But, to answer your question, no, most calculus teachers do not ask for theory on exams. Most give cookbook exams because otherwise most students would flunk. M.I.T. is not my alma mater. I flunked out, as did about half of the students in my day. Today, admission to M.I.T. is so highly prised that the effort to retain students has resulted in lowered standards. As of a few years back, M.I.T. started offering courses in "developmental math" ("developmental" is math ed jargon for "remedial"). But "math ed", a very important area in education, is not the same as "math". 4) I would say that one of the branches of abstract algebra is vector spaces, that at an elementary level, the study of vector spaces are usually called linear algebra, and that the study of determinants is a branch of linear algebra, and that the study of SL(2) is a branch of the study of determinants. Of course, the study of SL(2) is also a branch of group theory, and here the tree analogy breaks down, since in some sense all mathematics is one. But to the layperson, the picture of a branching subject is useful, and corresponds roughly to the list of prerequisites for undergraduate math courses.

I do understand your point -- that the way calculus is taught today in the US it is more a set of rules and not really mathematics at all. This is a valid criticism. But the only important point in all this is that standard sources describe calculus as a branch of mathematics. We should too. Rick Norwood (talk) 14:09, 28 November 2007 (UTC)

I guess after thinking and reading more, I have to say the word branch is misleading. It's not a proper branch anymore. I agree field is better and covers the important part of what the word branch is used for anyway. It's unfortunate that some sources use it that way, but using that word just because they use it in their introduction isn't an entirely proper way to use sources anyway. Unless the central point of the source is to describe whether calculus is a proper branch or not, then the choice of wording used in the source isn't definitive. More specifically to what the intro should say, I'm not opposed to what Rob has said "Calculus was an important branch of mathematics for hundreds of years and remains a part of mathematics as a subset of the modern branch of Analysis." being part of the intro, but it tells what it was not what it is. It is currently an important part of math education (and nearly all sciences by extension) and it is a field of mathematics, a subset of analysis. I think we need to get that accross first. - Taxman ^Talk 14:56, 28 November 2007 (UTC)

Another "standard" encyclopedic sources, the 1911 Brittanica, as rendered by LoveToKnow:[1]

INFINITESIMAL CALCULUS. 1. The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis.

Personally I'd say that Calculus is a tool, comprising a collection of notations and methods initially invented independently by Leibniz and Newton, which is very useful in tackling some, but not all, problems studied in Analysis. Consider the function f defined by f(x) = 0 for irrational x and f(p/q) = q⁻² for rational arguments given in simplest terms. When we prove that this function is continuous and has a vanishing derivative on the irrationals, we don't use any of the methods of the Calculus, which are useless here. Yet this is clearly the province of Analysis. On the other hand, when we apply the chain rule, or do integration by parts, we are wielding the tools of Calculus.  --Lambiam 16:19, 28 November 2007 (UTC)

I agree with what Taxman and Lambiam have just said. On Taxman's comments, indeed my suggestion "Calculus was a branch..." isn't as good as it could be and I don't think it should be the first sentence, I just think the idea of what it says should appear in the intro somewhere. I also agree with Rick Norwood's response to my 4th question, as determinants are a branch of linear algebra, one could probably say "Calculus is a branch of analysis" (hey perhaps the article could start "In mathematics, calculus is a branch of analysis" and go on to say that for over 150(?) years it was its own branch of mathematics).

To respond to Rick Norwood's other points, firstly Royden's book, in the words of the publisher is the "classic introductory graduate text", so yes it probably assumes prior knowledge of theorems on continuous functions. Secondly, my point is not that calculus is poorly taught, I in fact think that calculus is closer to what Lambiam above has said. To support this, consider the fact that calculus was indeed invented in the late 1600's and was only formalised in the 1800's (by Weierstrass, Cauchy, Riemann, etc.) and that during the period between these times, calculus was indeed a set of rules (L'Hospital's Rule, the chain rule) and methods (Newton's book Method of Fluxions) to solve certain problems. The proofs given when one covers calculus theoretically in an undergraduate course are from the 1800s, for example, one uses the Riemann integral, which did not exist before the 1840s. I may go so far as to say that, yes, certain calculus courses incorporate some proofs, but that these proofs are from analysis, not calculus, but this isn't we need to discuss right now, nor put in this article.

In summary, I think something like "In mathematics, calculus is a branch of analysis" or "In mathematics, calculus is a field of analysis" is more appropriate. I also think that it should be said that calculus was a branch of mathematics in its own right for a long time, and that it then became a subset of analysis. RobHar (talk) 01:12, 29 November 2007 (UTC)

New stuff about Archimedes and Calculus

http://www.sciencenews.org/articles/20071006/mathtrek.asp Gwen Gale 09:17, 8 October 2007 (UTC)

Summary: Heretofore unknown texts --> Early conception of limiting values by Archimedes used in calculations (parabola triangle example)+ distinction between "actual" infinity vs. "potential" infinity for calculations. Archimedes worked with both. The former only in his lost work.

—Cronholm¹⁴⁴ 09:40, 8 October 2007 (UTC)

Excellent article. There should definitely be a sentence about it here, and a paragraph in History of Calculus. Do you want to write it, or shall I? Rick Norwood 13:01, 8 October 2007 (UTC)

You go ahead, I have a term paper to write. :) —Cronholm¹⁴⁴ 03:03, 9 October 2007 (UTC)

Calculus of Antiquity

Ideas of calculus were developed earlier, in Egypt...

Sorry I had to delete Egypt. There has NEVER been any evidence that the Egyptians had developed any for of calculus whatsoever. We do however have the Moscow Papyrus which shows that the Egyptians had correct calculations for complicated volumes such as the frustum of a pyramid, but NEVER have we been given any evidence of them developing calculus.

I hope others can follow suit and check the claims made to other parts of the world. And when claims have been verified please find appropriate dates for the developments and cite resources. --123.100.92.83 19:18, 19 October 2007 (UTC)

There are many such claims, and they are hard to check. We really need someone who reads the language to check such claims, but how many Wikipedians read Sanscrit? Rick Norwood 12:51, 21 October 2007 (UTC)

The pump don't work 'cause the vandals took the handle.

Good work fighting vandalism, Gscshoyru. Rick Norwood 12:53, 21 October 2007 (UTC)

History

The History section is a bit long. I don't think that it is worth mentioning every time that the area of a circle was determined. The fact that Cauchy and Riemann only get a passing mention, and 3 different people are mentioned for calculating a circle's area at different points in history is sort of rediculous. I would suggest that the history section be trimmed down, to make mention of the fact that similar concepts had been developed before Newton and Leibniz.(Lucas(CA) (talk) 06:09, 16 December 2007 (UTC))

Calculus was not developed in India. At the very least, just find any Calculus textbook, go to the index, and look up the name Aryabhatta or any of the names I am removing. Their names are not in any index. And for good reason: they did not develop calculus. Aryabhata's contributions belong in the geometry article. Calculus relates areas of functions to their antiderivatives. Aryabhatta nowhere relates the idea of antiderivative (which had not been developed by his time) to finding the area under a curve. The method of exhaustion is a geometric technique, it is not calculus. Look at the first and second "fundamental theorems of calculus" for proof of this. They relate integrals to their antiderivatives. This connection was not known until the 17th century.

Every calculus text on the market today mentions nothing about Aryabhata (and they go to lengths to list the names who made important contributions to the development of calculus. Names like Wallis, Lagrange, Rolle, de Fermat etc that were mentioned in the previous wiki pages.) This goes for the textbooks by Larson/Hostetler, Stewart, and Thomas'.

Furthermore, look at the work cited to support this person's claim. Do a control-f search on that page for 'calculus'. Nothing comes up, again because Aryabhatta did not contribute to this field. So why is he mentioned in a page about calculus? That is why I am removing his name and picture. If someone wants, they can add him to the page about geometry or algebra, this page is about calculus. These distinctions are important if we want to keep things organized. —Preceding unsigned comment added by 70.185.199.182 (talk) 14:30, 9 May 2008 (UTC)

You have removed an entire paragraph of non-Western contributions. I agree with your assessment that Aryabhata should be removed, you also deleted references to Indian and Islamic mathematicians who are relevant to the development of calculus. For instance, the Kerala school is widely recognized as having made important contributions to the ideas of calculus in non-Western civilization. silly rabbit (talk) 15:40, 9 May 2008 (UTC)

Terminology

About half-way through the section on the Development of calculus is the following sentence. "Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhazen) was the first to derive the formula for the sum of the fourth powers, and using mathematical induction, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus." I think the integral link should actually link to Integer but I am not certain of this as it looks like both uses of the term integral are being used here. Also I think "powers" right after this usage of integral should be singular, I think an argument can be made either way. -AndrewBuck (talk) 04:10, 10 February 2008 (UTC)

Yes, it should link to integer (thanks). No, I think it is supposed to be plural. Would someone care to comment on what the "sum of integral powers" actually means? The most obvious meaning is the Riemann zeta ζ(s) for s an integer, but that's not exactly clear from the text. 04:17, 10 February 2008 (UTC)

I looked it up. I must be tired indeed, for the ibn al-Haytham formula is just that the sum of the first n fourth powers

$\sum _{k=1}^{n}k^{4}=\left({\frac {n}{5}}+{\frac {1}{5}}\right)n\left(n+{\frac {1}{2}}\right)\left((n+1)n-{\frac {1}{3}}\right).$

I'm not sure what it means by the sum of any integral powers. al-Haytham almost assuredly did not possess a general formula. Silly rabbit (talk) 04:46, 10 February 2008 (UTC)

Learning Calculus Faster

I really do appreciate all the information about this article, but it seems like their would be an faster way to learn this. Maybe just a list of formulas explaining step by step solutions. I have ADD so it makes it hard to read all of this information. Does anyone have any non ignorant ideas? —Preceding unsigned comment added by 76.126.198.46 (talk) 22:28, 25 February 2008 (UTC)

Like King Ptolemy I before you, you are asking for the Royal Road. If you don't understand the concepts behind the formulas well, you will have a hard time applying them appropriately. In the end, the investment in getting the basis right will pay off.  --Lambiam 11:08, 26 February 2008 (UTC)

Haha, thanks. —Preceding unsigned comment added by 76.126.198.46 (talk) 04:08, 1 March 2008 (UTC)

Problems solved in detail

If you have the smarts, can you type out what you do while solving 'any' of the problems in this article? The pictures are great, but i am left wondering what steps you took to get from A to B. Pretty sure most of it's just algebra; but i think if you do this it will help new calculus students.

Best Regards, The Nate DIZZLE

 —Preceding unsigned comment added by 76.126.198.46 (talk) 03:02, 12 March 2008 (UTC) 

This is not a calculus textbook and isn't intended to teach beginning students how to do problems in detail. I suggest you find a calculus textbook and work through the problems there to learn the process if you are having difficulty. PhySusie (talk) 10:53, 12 March 2008 (UTC)

Smooth infinitesimal analysis

I don't like the sentences concerning smooth infinitesimal analysis that have been added recently, like:

Another alternative is smooth infinitesimal analysis in which calculus is based on the concepts of microstraightness (of continuous functions) and of nilsquare infinitesimals. The logic of smooth infinitesimal analysis is intuitionistic logic which differs from classical logic in that it does not include the Axiom of Choice.

I feel that smooth infinitesimal calculus is used even less often than nonstandard analysis, and thus it should also get less space in this article. In other words, we only mention that it exists and let the reader go to the article for any further details. I amended this article accordingly.

Incidentally, I would replace the Axiom of Choice with the law of excluded middle in the above fragment. -- Jitse Niesen (talk) 14:16, 29 March 2008 (UTC)

I agree, WP:NPOV's undue weight section says that we can only give ideas space in articles in their relation to their importance to the topic. Unless reliable sources can be found that show that smooth infitesimal analysis is used extensively and is has widely impacted the use and teaching of calculus it should not get much or perhaps even any space in this article depending on how much impact can be shown to exist for it. This is the main calculus article and only the main topics can be given space. - Taxman ^Talk 16:48, 29 March 2008 (UTC)

It is not clear what the meaning is of "being used" with relation to smooth infinitesimal analysis. Like constructive mathematics, it is in some ways less powerful (and in other ways more powerful) than classical analysis. It is clearly more elegant than Bishop's Foundations of Constructive Analysis. Inasmuch as it may be viewed as an alternative, it is not in the calculus aspect (in the sense of a body of form-based rules), but in the foundational aspect. Can you measure how much a foundational approach is "used"? The people who have published on this are not the least among category theorists. With due respect to the undue-weight clause, I think it can at least be mentioned. The logic you can reconstruct from the approach goes beyond the propositional logic described in our article on intuitionistic logic and rejects both LEM and AC, but gives you back that all functions defined on R are continuous, a theorem of Brouwer.  --Lambiam 09:07, 30 March 2008 (UTC)

If you can argue for a different metric than "use" I'm fine with that, but there has to be some demonstrable bar of importance that it meets in order to justify mention at all. I probably just should have said generic importance in the first place. So basically you are saying you think there is more than enough published on this topic by people of importance to justify inclusion? Can you pull up anything that would show evidence to justify inclusion in the main calculus article? - Taxman ^Talk 17:11, 30 March 2008 (UTC)

I agree with Taxman that some specific reference linking smooth infinitesimal analysis and calculus should be found. At the very least, this will help to ascertain the extent to which it should be mentioned here. I have reverted 82...'s most recent edit [2] as it clearly places undue weight on smooth infinitesimal analysis (limits are no longer the standard approach to calculus? huh?) In this case, I recommend that we stick to reliable sources as much as possible. silly rabbit (talk) 15:44, 31 March 2008 (UTC)

Some minor edits

I agree with a previous poster that the history section does indeed go into sometimes unnecessary detail, and the overuse of names, places, dates and texts make the section hard to navigate and read. The Limits section only has a few sentences on the current use of limits (the first paragraph is mostly about the use of infinitesimals), and since limits are an important part of calculus, I think an example would greatly benefit the section. Though there is a 'limits' article on wikipedia that goes into much more detail, I think it's impossible to get a real feel for what limits are by just reading that short paragraph - one would be forced to go into the limits article to understand what they are. Notation should be added as well as one example, and this wouldn't add too much to the length of the article. In the derivatives section, there is an example shown where F(x)=x^2 and F'(x)=2x. Wouldn't it make more sense to explain that the derivative of x^n is nx^(n-1)? It gives a clearer understanding of how a derivative is obtained, and we could leave the previous example for explanation if need be. I would also suggest an equation for the antiderivative section as well. Finally, a very minor suggestion - when providing the equation "distance = speed x time", there is no need to have the equation written out like that, and formatted similarly to the other equations in the article - it makes it seem like that equation is a calculus term, and is unnecessarily formatted. Nessalora (talk) 03:41, 27 May 2008 (UTC)

I disagree with adding more about limits for two reasons: the main article on limits is very prominently linked, and there is an explicit calculation involving limits which gives a flavor of what elementary calculations in calculus are like. It is one of the beauties of an electronic encyclopedia that a quick check to find out what limits are and then returning to the main article to read more are both very easy. I don't know exactly what you mean about adding notation, but notation for limits is certainly covered in this article in the examples given.

Consensus against replacing the computation of the derivative of x² with that of xⁿ was reached on this talk page at Polynomial calculations above, and for good reasons: the current example is more concrete and therefore easier (at least in this case) to understand.Xantharius (talk) 18:02, 27 May 2008 (UTC)

Fundamental Theorem notes

As a somebody who took calculus in high school 20+ years ago, I think the article is useful and understandable quick summary, with the exception only of this part of the Fundamental theorem section:

"Furthermore, for every x in the interval (a, b),

${\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).$

I don't think a reader who doesn't already understand calculus will be able to meaningfully parse this equation.

Earlier in the article it is suggested the "dx" means "with respect to x. So what does "d/dx" mean? Further who or what is "t" and what does it have to do with the equation?

Really quite curious about this, since apparently the equation is "key to massive proliferation of analytic results"

Joshk6 (talk) 05:32, 20 June 2008 (UTC)

That notation is indeed not explained in this article, and therefore it should perhaps not be used. For the notation, see Leibniz's notation#Leibniz's notation for differentiation. A possible replacement text:

Conversely, the function F defined as a definite integral of ƒ, as follows,

$F(x)=\int _{a}^{x}f(t)\,dt\,,$

has ƒ as its derivative for every x in the interval (a, b):

$F'(x)=f(x)\,.$

 --Lambiam 22:20, 26 June 2008 (UTC)

IMO the Liebniz notation is so important, and so widely used, that this article absolutely needs to explain it. I've added an explanation of it, and I've made a section about it below on the talk page.--76.167.77.165 (talk) 18:53, 7 March 2009 (UTC)

I'm happy to see the addition of an explanation of Liebniz notation...but it could be a little more. Nevertheless, I applaud the addition thus far Gingermint (talk) 03:04, 16 June 2009 (UTC)

Wouldn't it also be proper to include a link to the main article Leibniz's notation? I would think so, but would like other opinions. ---JamesWill (talk) 08:20, 7 July 2009 (UTC)

I'm in the same position as Joshk6 and would really appreciate someone who understands calculus well expanding this section, continuing with the previous example of x squared and its derivative 2x. It appears to me at this point you're finding the area under 2x, which is linear. How do you calculate the antiderivative of x squared? And what is "t"? That just comes out of nowhere. I found this article absolutely excellent for my needs right up to this point, where I'm left in a cloud of confusion ---- TomW

Topics in Calulus

I suggest that the following articles be merged

• Shell method

• Disc method

• solid of revolution --this one is a good one I think it has potential to be an A-class Article but it is not even mentioned in this article

I also suggest

that Arc length be added to topics in Calculus

I just want to hear what you all think --GlasGhost (talk) 02:32, 1 August 2008 (UTC)

Are you suggesting that these articles be merged here? That doesn't seem like a very good idea. The subject of calculus is quite large, and we cannot hope to accommodate all of it in the article. There is a List of calculus topics for other topics, to which I will add your suggestions. siℓℓy rabbit (talk) 03:15, 1 August 2008 (UTC)

thx I was looking for that article but couldn't find it--GlasGhost (talk) 22:12, 16 August 2008 (UTC)

Where are the Applications?

The Applications section of the Calculus page of Wikipedia states-

"Calculus is used in every branch of the physical sciences, in computer science, statistics, engineering, economics, business, medicine, and in other fields."

Wikipedia then gives between 7-10 examples, some of them pretty weak especially in the light of the claim that it is used in seven different major fields, one of which is "every branch of the physical sciences". (call them astronomy, physics, chemistry meteorology) which is a total of 10 major disciplines. 7 examples. We (I) do not seem to be proving the point that calculus can be applied to all of these disciplines.

Is calculus the BASIS for a lot of formulas but not really USED for calculating answers to problems on a day to day basis? Is that the problem. If so, I think that is an important distinction that should be emphasized.

Please allow me to digress for a paragraph. I am a government engineer who passed the calculus sequence 21 years ago and I don't feel I really know the answer to the question "How is calculus applied". I did apply one example to Wikipedia (The planimeter example) but otherwise I can't answer the question concerning how calculus is applied. I will throw in the caveat that I work for the government and they hire a bunch of engineers and then have us do mostly non engineering work.

Ultimately I want others to show me examples even though I can not do so myself. And I realize the hypocrasy (sp?) in that statement.

This is my first Wikipedia edit. So be gentle and guiding in your statements. —Preceding unsigned comment added by Tfeeney65 (talk • contribs) 21:07, 17 October 2008 (UTC)

I believe this deserves further discussion. As it would be unwise to list applications for every branch of science and statistics listed in the article, perhaps creating a page entitled Applications of Calculus would be the best course? Otherwise it seems this section is lacking and only adding to the not-up-to-par standard of the article. ---JamesWill (talk) 08:28, 7 July 2009 (UTC)

leaves to be desired

The following section has room for improvement if anyone would like to take a look at it: "Limits and infinitesimals Main articles: Limit (mathematics) and Infinitesimal Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". On a number line, these would be locations which are not zero, but which have zero distance from zero. No non-zero number is an infinitesimal, because its distance from zero is positive. Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals do not satisfy the Archimedean property. From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals. This viewpoint fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.

In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use ordinary numbers. From this viewpoint, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are easy to put on rigorous foundations, and for this reason they are usually considered to be the standard approach to calculus." Katzmik (talk) 16:03, 15 January 2009 (UTC)

It doesn't look bad to me. What is it that you don't like about it? The only minor quibble I'd have is that it jumps back and forth a little chronologically.--76.167.77.165 (talk) 17:56, 7 March 2009 (UTC)

errors in section on Limits and infinitesimals

The section "Limits and infinitesimals" had some problems, in my opinion, which I've tried to fix. It stated "On a number line, these would be locations which are not zero, but which have zero distance from zero." This was incorrect. In nonstandard analysis, for example, an infinitesimal does not have zero distance from zero; the uniqueness of zero is a fact that can be expressed in first-order logic, so the transfer principle says that zero is unique on the hyperreals as well. This statement was also sort of a muddle: "No non-zero number is an infinitesimal, because its distance from zero is positive." The mistake is similar to the mistake in the first statement. It seems as though the person who wrote these statements was trying to give clear distinction between reals and infinitesimals, and was also trying to explain why reals can't be infinitesimals. The first question boils down to the Archimedean principle. The second one depends on the foundational framework you're using for the reals, e.g., an axiomatic one that includes an axiom of completeness, or a constructive one such as Dedekind cuts. I've made some edits to try to make this section correct, without making it too technical.--76.167.77.165 (talk) 18:18, 7 March 2009 (UTC)

Hmm. But when one is talking about distance, it sounds like the right framework is a more geometric and intuitive setting; like what one would get if one updated Euclid. I like your edits, though. They're definite improvements. (Though I'd like to point out that it's "Leibniz" not "Liebniz".) Ozob (talk) 18:29, 9 March 2009 (UTC)

Liebniz notation

The article conspicuously avoids the Liebniz notation when it introduces the derivative, and that's a perfectly reasonable choice to make. However, it then uses the Liebniz notation in other places, without explanation. I think the Liebniz notation is so widespread and useful that it really needs to be explained in the article. I'm going to add a little discussion of it.--76.167.77.165 (talk) 18:23, 7 March 2009 (UTC)

Rigorization

I disagree with the quote; "Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development.[11]" Calculus was rigourized by Baron Augustin-Louis Cauchy by 1828. No history of calculus I can find indicates any further work on calculus. The reference given to Unesco has no info on calculus or its continued development. 70.31.100.40 (talk) 01:29, 2 November 2009 (UTC)

The problem arises because the study of the derivative and integral and their properties, called "calculus" at the undergraduate level, is called "analysis" at the research level. Rick Norwood (talk) 13:08, 25 August 2010 (UTC)

Misuse of sources

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the cleanup page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed. Tobby72 (talk) 21:42, 24 August 2010 (UTC)

This goes all the way back to 2007, and there are a large number of edits to check. I checked the last one, and it seems to accurately reflect a reliable source. Rick Norwood (talk) 13:10, 25 August 2010 (UTC)

Here is a typical misuse of sources: In this edit Jagged introduced the text which now reads "the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus" with this source. But the source merely mentions that Egyptians knew a certain formula, and it speculates (in a student exercise) that dividing a volume into small blocks might have led to the formula. The statement in the article should be removed. I sampled Jagged's edits and they appear to be connected with Calculus#Ancient and Calculus#Medieval where Jagged introduced the text on Ibn al-Haytham ("was the first to derive the formula") and much more. To fix the mess, we need to read the sources with a skeptical eye while checking each claim in the Ancient/Medieval sections (there is no need to check each of Jagged's edits). I would remove anything that looks somewhat dubious in those sections (if really unsure, the removed text could be copied to here). In particular, when Jagged used the word "first", it is usually wrong (or fails verification). Johnuniq (talk) 00:49, 26 August 2010 (UTC)

Everything under the Medieval section appears to check out. If anyone would like a copy of the unlinked sources, contact me. Anyone that finds other discrepancies should post them here. Hexagon70 (talk) 00:29, 10 October 2010 (UTC)

In Our Time

The BBC programme In Our Time presented by Melvyn Bragg has an episode which may be about this subject (if not moving this note to the appropriate talk page earns cookies). You can add it to "External links" by pasting * {{In Our Time|Calculus|b00mrfwq}}. Rich Farmbrough, 03:01, 16 September 2010 (UTC).

Maria Gaetana Agnesi

I just reverted this edit by Lorynote which added:

File:Maria Gaetana Agnesi.jpg

Maria Gaetana Agnesi' was an Italian linguist, mathematician, and philosopher. Agnesi is credited with writing the first book discussing both differential and integral calculus. The plane curve, known as versiera, is also called the "Witch of Agnesi" [ref]Agnesi Witch Agne Scot, Agnesi.

My edit summary mention "unsourced". I see that is not correct since the refs do offer some support for the statement (since the refs were on the second sentence I jumped to the conclusion that the only support was Maria Gaetana Agnesi). However, the material definitely should not be in the "Significance" section, and there needs to be consideration of whether the material is WP:DUE (did the book describe as in a text book, or did it develop?). Johnuniq (talk) 00:29, 4 December 2010 (UTC)

The fact is that she is a pioneer, a major figure for the history of maths and calculus. It answer the question "who wrote th first book on calculus?". I see this as a fundamental information. Lorynote (talk) 10:35, 4 December 2010 (UTC)

From About.com: Women History, Agnesi; And: Belle vue college, Agnesi. Lorynote (talk) 11:00, 4 December 2010 (UTC)

This source says that "her two volume textbook was the first comprehensive textbook on the calculus after L'Hopital's earlier book", so L'Hopital would be more fundamental, yet is not mentioned in the article. The other source says that "it was one of the first and most complete works on finite and infinitesimal analysis.". So there is no support for "Agnesi is credited with writing the first book...", and neither here, whereas this source is "Condensed from "The Pioneering Women Mathematicians" by G.J. Tee in The Mathematical Intelligencer", so not really a reliabe source. The sources do mention the Witch of Agnesi, but that is more on-topic in analytic geometry, not in calculus. DVdm (talk) 11:10, 4 December 2010 (UTC)

Ok, so one can say she wrote first comprehensive book. Lorynote (talk) 12:07, 4 December 2010 (UTC)

No, we can not say that. She wrote the first comprehensive book after L'Hopital's earlier book, and there is no reason to mention that, since L'Hopital is not mentioned in the article either. DVdm (talk) 12:49, 4 December 2010 (UTC)

Although there are two sources to support that she was the pioneer and only one to support the after someone else we can keep the second; mentioning she wrote one of the first books. Besides, the source clearly cites his book and her compreheensive book. The fact L´Hopital is not mentioned here it´s not relevant; once his name is on WP anyway and the link will not be red. Lorynote (talk) 13:15, 4 December 2010 (UTC)

Also note that the first source says in its disclaimer on http://jwilson.coe.uga.edu/ : "The content and opinions expressed on this Web page do not necessarily reflect the views or nor they endorsed by the University of Georgia or the University System of Georgia.". This is someone's personal web site, so It cannot be taken as a wp:reliable source either.

I think it would be okay to use the second source to write this short statement in the article:

One of the first and most complete works on finite and infinitesimal analysis was written In 1748 by Maria Gaetana Agnesi.

No details about or picture of Agnesi are needed here. Readers who click the link get all they want. What do the other contributors think about this? DVdm (talk) 13:42, 4 December 2010 (UTC)

You can guess my opinion; I vote for a picture and more details about her (just the very short 'she was italian, mathematician, philosopher'). My very honest opinion is that she has a brilliant work and she was recognized as such. Lorynote (talk) 14:13, 4 December 2010 (UTC)

I agree it needs to be short, no bio needed (no-one else has one), but a mention needs to be there. The Agnes Scott College source is adequate - there are several mathematical sources on the Witch of Agnesi curve - I think as well as your sentence, one sentence on the Witch (which is very well known and has its own article, and there's no contention that it was studied by and named after Agnesi) would be adequate. People can read both articles for more info, but if we don't put a mention in, people won't know to look. This is always the challenge of an article like this.

One of the first and most complete works on finite and infinitesimal analysis was written in 1748 by Maria Gaetana Agnesi. The Witch of Agnesi curve is named after Agnesi, who wrote about it in 1748 in her book Istituzioni Analitiche. (this citation taken from the Witch article)

Picture I'm in favour of. We have Newton and Leibzitz - the fathers of calculus (that sounds odd I know). Agnesi I agree is probably not more notable as a mathematician (as opposed to a female mathematician) than several others who advanced one area of calculus or other, but I think the addition of her picture would possibly attract the interest of more readers, because she is a woman. It's a bit tokenistic I know, but pictures should mho pique the interest, not just decorate the page. Elen of the Roads (talk) 16:27, 4 December 2010 (UTC)

Elen, I would sign next to your words as if they were mine. Lorynote (talk) 16:34, 4 December 2010 (UTC)

I have no problem with Agnesi being mentioned, but shouldn't there be a sentence about l'Hôpital as well, since he actually produced the first textbook (plus l'Hôpital's rule of course)? Favonian (talk) 16:43, 4 December 2010 (UTC)

I´m for L´Hôpital as well! Sure, he´s welcome! Lorynote (talk) 17:15, 4 December 2010 (UTC)