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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)
the 'applications' section uses the word calculus too much. Am I crazy?
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 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)
Isn't calculus often called "The Calculus"? Zginder 23:43, 7 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)
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)
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)
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)
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)
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)
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)
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)
Another "standard" encyclopedic sources, the 1911 Brittanica, as rendered by LoveToKnow: [1]
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)
http://www.sciencenews.org/articles/20071006/mathtrek.asp Gwen Gale 09:17, 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. :) — Cronholm 144 03:03, 9 October 2007 (UTC)
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)
Good work fighting vandalism, Gscshoyru. Rick Norwood 12:53, 21 October 2007 (UTC)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 | Archive 6 | Archive 7 | Archive 8 |
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 | Archive 6 | Archive 7 | Archive 8 |
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)
the 'applications' section uses the word calculus too much. Am I crazy?
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 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)
Isn't calculus often called "The Calculus"? Zginder 23:43, 7 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)
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)
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)
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)
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)
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)
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)
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)
Another "standard" encyclopedic sources, the 1911 Brittanica, as rendered by LoveToKnow: [1]
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)
http://www.sciencenews.org/articles/20071006/mathtrek.asp Gwen Gale 09:17, 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. :) — Cronholm 144 03:03, 9 October 2007 (UTC)
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)
Good work fighting vandalism, Gscshoyru. Rick Norwood 12:53, 21 October 2007 (UTC)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 | Archive 6 | Archive 7 | Archive 8 |