![]() | 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 2 | Archive 3 | Archive 4 | Archive 5 |
The comment(s) below were originally left at Talk:Integral/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
* This article needs to be expanded to cover the topic more broadly. Integration is one of the key mathematics articles, and it should provide a good balanced coverage of the concept. Article with this name should not be reserved for only Integral of real-valued functions one real variable. See talk page for more suggestions. Stca74 17:24, 15 May 2007 (UTC) |
Last edited at 11:39, 2 June 2007 (UTC). Substituted at 14:32, 14 April 2016 (UTC)
Shouldn't we include a proof that integrals can be calculated through antidifferentialtion?
I'll see if I can write one —The preceding unsigned comment was added by Sav chris13 ( talk • contribs) 05:10, 5 February 2007 (UTC).
Uh... I guess this was written before the fundamental theorem was in the article.-- Cronholm144 12:06, 2 June 2007 (UTC)
Let f(x) be a function
and F(x) [the integral] be the function of the area under f(x)
Also note the relationships:
Area = Length X Width
Gradient = Rise / Run
The Rise in F(x) is F(x + h) - F(x) as h --> 0
The Run is h
The derivative of F being repressented by
[F(x + h) - F(x)]/h as h --> 0
Now F(x) is the Area function
And values along the x-axis represent the "width" of our area
So h is the width of this area
So
Gradient = Rise / Run = Area / Width = Length
Ie: [F(x + h) - F(x)]/h = f(c)
For some value of c which is between (x + h) and x
Now as h --> 0 c will approach x
[F(x + h) - F(x)]/h = f(x) as h --> 0
Hence the relationship between F and f is
F'(x) = f(x) —The preceding unsigned comment was added by Sav chris13 ( talk • contribs) 07:19, 5 February 2007 (UTC).
Are there any thoughts about this post? if not, I will archive it.-- Cronholm144 12:08, 2 June 2007 (UTC)
You should say that this was discovered by Newton, and the others had some different proves (like Leibniz, and we should put their proves, I'll try to write Leibniz's, and Cauchy's proof). Some things you should change in your proof like Rise/Run, Area... You should put some graphic explanation. I'll do another version soon.
I'll log in later as CRORaf 195.29.73.31 11:03, 9 June 2007 (UTC)
I linked to this article trying to explain what I meant by "integrating power with respect to time to get energy" but was horrified to find that within the first 12 lines of text we were already using set theory notation and Rieman definitions - before we even hit the table of contents. I strongly suspect someone who really understood the topic could explain it for the proverbial bright 12-year-old reader before disappearing into hihger maths; the basic concept deserves a more lucid explantion than this. I'm NOT a mathemetician but if someone doesn't come along and write a lucid introduction within the next few days, I *WILL* haul out my old maths books and write a better one. Painting a picket fence would be a good introduction to the topic - make the fence height variable, and make the pickets smaller and smaller...how much paint do you need? That sort of homey explantion before we get into the runes. -- Wtshymanski 00:19, 9 March 2007 (UTC)
There is an anonymous user who is replacing italic d's with upright d's (for differentials, or the exterior derivative) in many articles. This is a point of view which I support. Both usages are common, the italic d being more common in the US, and the upright (roman) d being more common in the UK. As I am from the UK, my point of view may be biased (although in general I favour US spellings for math articles, especially fiber). However, I think the upright d works particularly well in wikipedia because of the unique mixture of math and wiki text in which it occurs. I therefore not only presume (as we all should) that this user is acting in good faith, but think this good faith is justified. Geometry guy 23:11, 22 March 2007 (UTC)
I think we should consider including the definition of the integral in this article. That is,
the lim N->infinity of the summation from k=1 to N of f(a-kΔx)Δx, where Δx=(b-a)/N. This is describing the method of using an (approaching) infinite number of rectangles to produce the area of the function f in the interval [a,b].
This is in relation to integration on 2D planes. I am well aware that there are refined definitions for integrals of other circumstances, which should also be included. The reason I bring this up is because we include the definition of the derivative under its own section but only give the fundamental theorem of calculus under the Integration section. I feel this definition should be included.
If we could, I would like to discuss this and if others want, I could spearhead the initiative myself(of course, with the help of others).
Gagueci 20:39, 1 May 2007 (UTC)
Sorry about the multiple posts, my message was not going through (so I thought) so I clicked save changes a few times. When I realized later that three had been posted, I deleted the other two. Gagueci 20:11, 3 May 2007 (UTC)
As outlined in my rating comment, I think the scope of this article needs to be broadened to cover the concept of integral in appropriate generality, not concentrating only on integrals of real-valued functions of one real variable. While this is a critical special case (and indeed the key building block for other inegrals), it is by no means sufficient for an article that aims to cover one of the most critically important mathematics concepts.
However, care should be taken not to introduce better coverage at the expense of too high level of demands for readers — this is likely one of the most viewed maths articles. Therefore most technical details belong to either in later sections or in particular separate articles.
Additions and changes proposed include:
I also agree that this article needs to have broader coverage. I gave its companion article Derivative a similar treatment a month or two ago. This is what it looked like before: it covered only one real variable, lacked balance, and had a number of organisational problems, just like this article now. One practical suggestion I can make is to make better use of (and improve) subarticles: in a core topic such as this, one cannot cover all aspects in sufficient detail in one article.
I generally agree with the above suggestions, although I think it is particularly important to keep the perspective as elementary as possible and to provide an overview: specific topics (such as the list of various integrals) should be approached here from the viewpoint of the general reader, rather than the specialist in integration.
The down-rating to B-class is entirely appropriate, and possibly even generous: this is still a long way from being a good article. In particular, while Loom91 does not support a more compact lead, I am afraid there is zero probability of promotion to GA status with the lead as it is: see WP:LEAD. However, I have found that it is a wasted effort to try to write a good lead while the body of the article is unsatisfactory (for one thing, the lead should, to some extent, reflect the content of the article). So I suggest efforts should be focused on improving the main part of the article. The lead will then (again, in my experience) fall much more easily into place. Geometry guy 19:20, 15 May 2007 (UTC)
I think the article should remain simple. It used to have a comparison or Riemann and Lebesgue integration, and perhaps other stuff (I wrote a lot of that). Someone else took it out and over all I think that was a good move. It would be much better to explain the simplest concept of integration as well as possible and perhaps flesh out the links to the other integration articles. I think integration of differential forms does not belong in this article. Also note that it is probably futile to attempt to cover integration in full generality. The Itô integral, or integration with spectral measures for instance, does not belong in this article.
Loisel 05:00, 16 May 2007 (UTC)
I hadn't seen the reworked derivative article when I wrote my original comment here, but should I have seen it, I would have pointed it out as a model to follow — it has just the type of broad coverage at accessible level with generous links to other (sub-) articles that I had in mind. As for Loom91's comment, I partially disagree: the goal of explaining "the simplest concept of integration as well as possible", if done at the expense of broad coverage, is more appropriate for an elementary textbook (for Wikibooks?) than an encyclopaedia article. That said, I am also in favour of devoting more space for the more elementary concept, provided that the reader is made aware of the bigger picture. As for differential forms (and / or its more "elementary" versions), I still think they deserve a paragraph or two, with surely the bulk of exposition in a separate article. And the same applies to stochastic integral as well: it surely needs at least a one-sentence mention (how did I forget that?). Spectral measures I see rather as an application (of general vector-valued integration) than as a new integration concept as such, and thus would briefly mention under vectorial integration and point at spectral measure. Stca74 05:43, 16 May 2007 (UTC)
I've noticed that some people write while others write . Is there a story behind these two different conventions? -- Itub 13:21, 21 May 2007 (UTC)
is an infinitesimal and can be treated as a normal variable; so both are the same and valid, but is much more common. More of a debate comes over v. . Dmbrown00 04:35, 31 May 2007 (UTC)
I don't think that both ways are good. is good, but means 195.29.69.197 09:35, 3 June 2007 (UTC)
Is this always true: ? -- Abdull 11:18, 24 May 2007 (UTC)
Yes. In general, (b-a) = -(a-b) Gagueci 19:07, 31 May 2007 (UTC)
In general a good article, but two points:
Dmbrown00 04:31, 31 May 2007 (UTC)
-- Cronholm144 12:12, 2 June 2007 (UTC)
This section must be elaborated on mathematically. Gagueci 19:11, 31 May 2007 (UTC)
Since the first paragraph (indeed, the first sentence) frightens me, and the rest of the article needs massive work as well, I have concentrated my initial efforts on non-creative writing. Specifically, I have laboriously searched the Web and added some great references for the " History of integral notation" section. In doing so, I have established a precedent that I wish to be followed: Harvard style references with automatic links. I have yet to properly templatize (!) the Leibniz citation, but that is a minor issue, which I will fix Real Soon. (I plead fatigue.)
I'm not yet concerned with naming names within the bulk of the article, because I expect that to happen as it is pummeled into submission. I see a need for better coverage of the basics (linearity!), but we should also touch on contour integrals, complex integration, measure theory and analysis, differential forms, and (if we're really brave/foolhardy) de Rham cohomology or some such. -- KSmrq T 12:29, 1 June 2007 (UTC)
I second that. Specifically, I am sad to say that this article screams "I was written by a mathematician," I think a copy-edit by an English person would be a great boon. Also, I was giving the article a full read-through and I got to the part about integrals with more than one variable... then the article ended. What happened? Where is the rest of the article? I am surprised how an article that seems well written when given a cursory glance can lack so very much. I agree with KSmrq's assessment about the basics, so let's get working!-- Cronholm144 11:35, 2 June 2007 (UTC)
P.S. I have created a sandbox User:Cronholm144/Integral for my more extreme edits. Anyone who wants to play is welcome.
The first paragraph is indeed appalling. It seems to have been written with an eye to generality, but devolves into verbose vagary to the point of being almost incomprehensible. I also note the full generality it seeks to cover is not actually dealt with in the article itself. In the meantime I've tried a rewrite of the first paragraph, which is the most glaring issue with the introduction. I would appreciate feedback, and hopefully we can make what I have even more accessible.
I still feel it is a little vague, and I'm dicing with the issue of smooth vs. continuous (smooth, in a general sense, is more accessible to laymen, but it has specific mathematical connotations which may be worth avoiding here). Suggestions and feedback are welcome. -- Leland McInnes 00:06, 3 June 2007 (UTC)
I've dropped it into place in the article for now. I'll start trying to clean up the rest of the introduction soon. -- Leland McInnes 17:08, 3 June 2007 (UTC)
I noticed that there is a sentence in the article that mentions that it is possible to prove that has no elementary antiderivative. While I'm sure that's true, the way the sentence reads begs the question of how to go about proving it. So it seems like there really ought to be a footnote citation that leads interested readers to an actual proof. (It may even be included in one of the references at the end of the article, but if so there's no indication which part of which reference to go to for the proof.) Dugwiki 20:51, 1 June 2007 (UTC)
Loom91 removed my introduction, citing the fact that a function does not need to be continuous to be integrable. I agree, but then that is not what my introduction said -- the claim is that the function should be defined on a continuous domain since a function defined on a discrete domain can simply be summed over a subset of the domain in the usual manner. Integration deals with the issue of extending this to continuous domains, and I feel this provides the most natural "intuitive" description of what integration achieves. I would hope that we could discuss the introduction instead of just reverting it -- if nothing else it is better than the existing introduction which makes almost no sense to anyone who isn't well schooled in what integration is already. -- Leland McInnes 21:07, 3 June 2007 (UTC)
Hey all, I have created a sandbox skeleton for the new look of the article and Leland has kindly outlined his vision for the article on the talk page. Please direct all major edits that you wish to try before adding them into the article proper there. Hopefully the skeleton can be given flesh in the sandbox and then life in the article proper. Cheers-- Cronholm 144 19:25, 4 June 2007 (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 2 | Archive 3 | Archive 4 | Archive 5 |
The comment(s) below were originally left at Talk:Integral/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
* This article needs to be expanded to cover the topic more broadly. Integration is one of the key mathematics articles, and it should provide a good balanced coverage of the concept. Article with this name should not be reserved for only Integral of real-valued functions one real variable. See talk page for more suggestions. Stca74 17:24, 15 May 2007 (UTC) |
Last edited at 11:39, 2 June 2007 (UTC). Substituted at 14:32, 14 April 2016 (UTC)
Shouldn't we include a proof that integrals can be calculated through antidifferentialtion?
I'll see if I can write one —The preceding unsigned comment was added by Sav chris13 ( talk • contribs) 05:10, 5 February 2007 (UTC).
Uh... I guess this was written before the fundamental theorem was in the article.-- Cronholm144 12:06, 2 June 2007 (UTC)
Let f(x) be a function
and F(x) [the integral] be the function of the area under f(x)
Also note the relationships:
Area = Length X Width
Gradient = Rise / Run
The Rise in F(x) is F(x + h) - F(x) as h --> 0
The Run is h
The derivative of F being repressented by
[F(x + h) - F(x)]/h as h --> 0
Now F(x) is the Area function
And values along the x-axis represent the "width" of our area
So h is the width of this area
So
Gradient = Rise / Run = Area / Width = Length
Ie: [F(x + h) - F(x)]/h = f(c)
For some value of c which is between (x + h) and x
Now as h --> 0 c will approach x
[F(x + h) - F(x)]/h = f(x) as h --> 0
Hence the relationship between F and f is
F'(x) = f(x) —The preceding unsigned comment was added by Sav chris13 ( talk • contribs) 07:19, 5 February 2007 (UTC).
Are there any thoughts about this post? if not, I will archive it.-- Cronholm144 12:08, 2 June 2007 (UTC)
You should say that this was discovered by Newton, and the others had some different proves (like Leibniz, and we should put their proves, I'll try to write Leibniz's, and Cauchy's proof). Some things you should change in your proof like Rise/Run, Area... You should put some graphic explanation. I'll do another version soon.
I'll log in later as CRORaf 195.29.73.31 11:03, 9 June 2007 (UTC)
I linked to this article trying to explain what I meant by "integrating power with respect to time to get energy" but was horrified to find that within the first 12 lines of text we were already using set theory notation and Rieman definitions - before we even hit the table of contents. I strongly suspect someone who really understood the topic could explain it for the proverbial bright 12-year-old reader before disappearing into hihger maths; the basic concept deserves a more lucid explantion than this. I'm NOT a mathemetician but if someone doesn't come along and write a lucid introduction within the next few days, I *WILL* haul out my old maths books and write a better one. Painting a picket fence would be a good introduction to the topic - make the fence height variable, and make the pickets smaller and smaller...how much paint do you need? That sort of homey explantion before we get into the runes. -- Wtshymanski 00:19, 9 March 2007 (UTC)
There is an anonymous user who is replacing italic d's with upright d's (for differentials, or the exterior derivative) in many articles. This is a point of view which I support. Both usages are common, the italic d being more common in the US, and the upright (roman) d being more common in the UK. As I am from the UK, my point of view may be biased (although in general I favour US spellings for math articles, especially fiber). However, I think the upright d works particularly well in wikipedia because of the unique mixture of math and wiki text in which it occurs. I therefore not only presume (as we all should) that this user is acting in good faith, but think this good faith is justified. Geometry guy 23:11, 22 March 2007 (UTC)
I think we should consider including the definition of the integral in this article. That is,
the lim N->infinity of the summation from k=1 to N of f(a-kΔx)Δx, where Δx=(b-a)/N. This is describing the method of using an (approaching) infinite number of rectangles to produce the area of the function f in the interval [a,b].
This is in relation to integration on 2D planes. I am well aware that there are refined definitions for integrals of other circumstances, which should also be included. The reason I bring this up is because we include the definition of the derivative under its own section but only give the fundamental theorem of calculus under the Integration section. I feel this definition should be included.
If we could, I would like to discuss this and if others want, I could spearhead the initiative myself(of course, with the help of others).
Gagueci 20:39, 1 May 2007 (UTC)
Sorry about the multiple posts, my message was not going through (so I thought) so I clicked save changes a few times. When I realized later that three had been posted, I deleted the other two. Gagueci 20:11, 3 May 2007 (UTC)
As outlined in my rating comment, I think the scope of this article needs to be broadened to cover the concept of integral in appropriate generality, not concentrating only on integrals of real-valued functions of one real variable. While this is a critical special case (and indeed the key building block for other inegrals), it is by no means sufficient for an article that aims to cover one of the most critically important mathematics concepts.
However, care should be taken not to introduce better coverage at the expense of too high level of demands for readers — this is likely one of the most viewed maths articles. Therefore most technical details belong to either in later sections or in particular separate articles.
Additions and changes proposed include:
I also agree that this article needs to have broader coverage. I gave its companion article Derivative a similar treatment a month or two ago. This is what it looked like before: it covered only one real variable, lacked balance, and had a number of organisational problems, just like this article now. One practical suggestion I can make is to make better use of (and improve) subarticles: in a core topic such as this, one cannot cover all aspects in sufficient detail in one article.
I generally agree with the above suggestions, although I think it is particularly important to keep the perspective as elementary as possible and to provide an overview: specific topics (such as the list of various integrals) should be approached here from the viewpoint of the general reader, rather than the specialist in integration.
The down-rating to B-class is entirely appropriate, and possibly even generous: this is still a long way from being a good article. In particular, while Loom91 does not support a more compact lead, I am afraid there is zero probability of promotion to GA status with the lead as it is: see WP:LEAD. However, I have found that it is a wasted effort to try to write a good lead while the body of the article is unsatisfactory (for one thing, the lead should, to some extent, reflect the content of the article). So I suggest efforts should be focused on improving the main part of the article. The lead will then (again, in my experience) fall much more easily into place. Geometry guy 19:20, 15 May 2007 (UTC)
I think the article should remain simple. It used to have a comparison or Riemann and Lebesgue integration, and perhaps other stuff (I wrote a lot of that). Someone else took it out and over all I think that was a good move. It would be much better to explain the simplest concept of integration as well as possible and perhaps flesh out the links to the other integration articles. I think integration of differential forms does not belong in this article. Also note that it is probably futile to attempt to cover integration in full generality. The Itô integral, or integration with spectral measures for instance, does not belong in this article.
Loisel 05:00, 16 May 2007 (UTC)
I hadn't seen the reworked derivative article when I wrote my original comment here, but should I have seen it, I would have pointed it out as a model to follow — it has just the type of broad coverage at accessible level with generous links to other (sub-) articles that I had in mind. As for Loom91's comment, I partially disagree: the goal of explaining "the simplest concept of integration as well as possible", if done at the expense of broad coverage, is more appropriate for an elementary textbook (for Wikibooks?) than an encyclopaedia article. That said, I am also in favour of devoting more space for the more elementary concept, provided that the reader is made aware of the bigger picture. As for differential forms (and / or its more "elementary" versions), I still think they deserve a paragraph or two, with surely the bulk of exposition in a separate article. And the same applies to stochastic integral as well: it surely needs at least a one-sentence mention (how did I forget that?). Spectral measures I see rather as an application (of general vector-valued integration) than as a new integration concept as such, and thus would briefly mention under vectorial integration and point at spectral measure. Stca74 05:43, 16 May 2007 (UTC)
I've noticed that some people write while others write . Is there a story behind these two different conventions? -- Itub 13:21, 21 May 2007 (UTC)
is an infinitesimal and can be treated as a normal variable; so both are the same and valid, but is much more common. More of a debate comes over v. . Dmbrown00 04:35, 31 May 2007 (UTC)
I don't think that both ways are good. is good, but means 195.29.69.197 09:35, 3 June 2007 (UTC)
Is this always true: ? -- Abdull 11:18, 24 May 2007 (UTC)
Yes. In general, (b-a) = -(a-b) Gagueci 19:07, 31 May 2007 (UTC)
In general a good article, but two points:
Dmbrown00 04:31, 31 May 2007 (UTC)
-- Cronholm144 12:12, 2 June 2007 (UTC)
This section must be elaborated on mathematically. Gagueci 19:11, 31 May 2007 (UTC)
Since the first paragraph (indeed, the first sentence) frightens me, and the rest of the article needs massive work as well, I have concentrated my initial efforts on non-creative writing. Specifically, I have laboriously searched the Web and added some great references for the " History of integral notation" section. In doing so, I have established a precedent that I wish to be followed: Harvard style references with automatic links. I have yet to properly templatize (!) the Leibniz citation, but that is a minor issue, which I will fix Real Soon. (I plead fatigue.)
I'm not yet concerned with naming names within the bulk of the article, because I expect that to happen as it is pummeled into submission. I see a need for better coverage of the basics (linearity!), but we should also touch on contour integrals, complex integration, measure theory and analysis, differential forms, and (if we're really brave/foolhardy) de Rham cohomology or some such. -- KSmrq T 12:29, 1 June 2007 (UTC)
I second that. Specifically, I am sad to say that this article screams "I was written by a mathematician," I think a copy-edit by an English person would be a great boon. Also, I was giving the article a full read-through and I got to the part about integrals with more than one variable... then the article ended. What happened? Where is the rest of the article? I am surprised how an article that seems well written when given a cursory glance can lack so very much. I agree with KSmrq's assessment about the basics, so let's get working!-- Cronholm144 11:35, 2 June 2007 (UTC)
P.S. I have created a sandbox User:Cronholm144/Integral for my more extreme edits. Anyone who wants to play is welcome.
The first paragraph is indeed appalling. It seems to have been written with an eye to generality, but devolves into verbose vagary to the point of being almost incomprehensible. I also note the full generality it seeks to cover is not actually dealt with in the article itself. In the meantime I've tried a rewrite of the first paragraph, which is the most glaring issue with the introduction. I would appreciate feedback, and hopefully we can make what I have even more accessible.
I still feel it is a little vague, and I'm dicing with the issue of smooth vs. continuous (smooth, in a general sense, is more accessible to laymen, but it has specific mathematical connotations which may be worth avoiding here). Suggestions and feedback are welcome. -- Leland McInnes 00:06, 3 June 2007 (UTC)
I've dropped it into place in the article for now. I'll start trying to clean up the rest of the introduction soon. -- Leland McInnes 17:08, 3 June 2007 (UTC)
I noticed that there is a sentence in the article that mentions that it is possible to prove that has no elementary antiderivative. While I'm sure that's true, the way the sentence reads begs the question of how to go about proving it. So it seems like there really ought to be a footnote citation that leads interested readers to an actual proof. (It may even be included in one of the references at the end of the article, but if so there's no indication which part of which reference to go to for the proof.) Dugwiki 20:51, 1 June 2007 (UTC)
Loom91 removed my introduction, citing the fact that a function does not need to be continuous to be integrable. I agree, but then that is not what my introduction said -- the claim is that the function should be defined on a continuous domain since a function defined on a discrete domain can simply be summed over a subset of the domain in the usual manner. Integration deals with the issue of extending this to continuous domains, and I feel this provides the most natural "intuitive" description of what integration achieves. I would hope that we could discuss the introduction instead of just reverting it -- if nothing else it is better than the existing introduction which makes almost no sense to anyone who isn't well schooled in what integration is already. -- Leland McInnes 21:07, 3 June 2007 (UTC)
Hey all, I have created a sandbox skeleton for the new look of the article and Leland has kindly outlined his vision for the article on the talk page. Please direct all major edits that you wish to try before adding them into the article proper there. Hopefully the skeleton can be given flesh in the sandbox and then life in the article proper. Cheers-- Cronholm 144 19:25, 4 June 2007 (UTC)