Talk:Line integral

Requested move

Just a question, why not put this under the title of "contour integral". I have a B.S. in physics and am currently working on my Ph.D., and in all my Math and Physics courses I have never heard this sort of integration refered to as a path integral, always as a contour integral. "path integral" is always used to refer to an integration over a set of paths in the sense of Feynmann. This may be a may vary based upon country of even university, but it's probably worth consideration.

In the several editions of vector calculus or multi-variable calculus textbooks my university uses (University of Arizona, current textbook in use ISBN 0-471-40952-9), it is invariably called a Path Integral. It sounds as though we may need a disambig page rather than a quick statement on the top of the article. I'd suggest moving Path integral to Contour integral and putting the disambig page at Path integral with links to Contour integral and Functional integration. Any other ideas? --Abqwildcat 00:40, 1 Jul 2004 (UTC)

I was rather taken back by your statement about your textbook. I thought it was fairly universal that calculus textbooks like the term "line integral", e.g. Thomas' calculus or Stewart. Anyway, I checked the book you cite and it uses line integrals. In fact, the chapter on it is called "Line integrals".

Additionally, in my experience, mathematicians do not use "path integral" for this concept, but prefer "contour integral" or "line integral". I've almost always heard "path integral" refer to integration in the Feynman sense.

Given the discussion so far, I believe "path integral" should be the title of the math physics method, with a disambig note at top with a link to "line integral" or "contour integral". --C S (Talk) 11:07, 29 May 2006 (UTC)

My experience is the same. I suggest moving to Line integral, which is more popular than Contour integral on Google and Google books; also, "contour integral" tends to carry a complex analysis connotation. And Path integral should simply redirect to Functional integration, with a dab note. Melchoir 22:21, 16 June 2006 (UTC)

To support my vote, I have consulted the classic and reputable text by Ahlfors, Complex Analysis, 3/e (ISBN 978-0-07-000657-7). At the beginning of chapter 4, Complex Integration, this concept is clearly named a line integral. Ergo, move. --KSmrq^T 21:38, 19 June 2006 (UTC)

Concur, move to Line integral. (I thought I had a copy of [Ahlfors], but I can't find it. — Arthur Rubin | (talk) 18:50, 20 June 2006 (UTC)

I moved the page (though I now wish I'd checked how many pages link to this page; it will take me some time to fix them up). -- Jitse Niesen (talk) 11:34, 21 June 2006 (UTC)

Why is it that after all this discussion, the (imho incorrect) term "path integral" is still used in the first sentence of the first paragraph? Pzinn (talk) 12:55, 10 November 2014 (UTC)

Brackets

Since when are brackets used for nested functions? RoboJesus 05:26, 16 August 2005 (UTC)

Where? Melchoir 22:42, 16 June 2006 (UTC)

paths and contours

I have a BS in math and am working on a PhD in math. The way I just learned it is that a contour is just a collection of smooth curves linked together. So I could have a path integral along the path which is a bottom half circle from 0 to 1 and then the line from 1 to i. Or I could call the whole thing a contour C and call the integral a contour integral along C. —The preceding unsigned comment was added by Tbsmith (talk • contribs) 02:54, 29 December 2005 (UTC)

vote

This is my website of contour integral example problems. Please someone add this link to the external links section of the main article if you think it's relevant and helpful.

http://www.exampleproblems.com/wiki/index.php/Complex_Variables#Complex_Integrals

—The preceding unsigned comment was added by Tbsmith (talk • contribs) 02:57, 29 December 2005 (UTC)

Equation understanding problems

Currently, you find this in the article

---

$\int _{\gamma }f(z)\,dz$

may be defined by subdividing the interval [a, b] into a = t₀ < t₁ < ... < t_n = b and considering the expression

$\sum _{1\leq k\leq n}f(\gamma (t_{k}))(\gamma (t_{k})-\gamma (t_{k-1})).$

---

May someone explain how we come from the first equation to the second one? --Abdull 10:47, 27 May 2006 (UTC)

The last factor, gamma-gamma, is a discrete version of dz. Melchoir 22:32, 16 June 2006 (UTC)

What is it?

I have to say that this article offers no basic explanation or definition as called for by WP:MSM. It states that it "is an integral where the function to be integrated is evaluated along a path or curve," which doesn't really offer an definition that would be understandable to someone who doesn't already know what it is, because it's too vague and ambiguous to really understand. I definitely think the introduction should be rewriten. He Who Is 18:13, 10 June 2006 (UTC)

Any suggestions, then? Perhaps a really clever picture? Melchoir 22:40, 16 June 2006 (UTC)

I tried a really clever picture. Its not super accurate, but I found a similar thing on a different page that really really made me understand the concept. Fresheneesz 19:51, 2 August 2006 (UTC)

Hey, not bad! I would suggest making the particle draw the broken path as it moves, but it's not a big deal. I do worry about the meaning, though. It seems like the integral being illustrated is one of the form

${\vec {V}}=\int {\vec {v}}\;ds,$

while the only two forms discussed in the article are

$F=\int f\;ds$ and $F=\int {\vec {v}}\cdot d{\vec {s}}.$

Thoughts? Melchoir 20:07, 2 August 2006 (UTC)

You're very welcome to modify the picture yourself (i'd suggest editing the higher quality, and firstly uploaded picture). Do you mean have the line be sequentually be created instead of only being shown at the end?

I'm still fuzzy myself on how a line integral works, but its my understanding that a line, and a vector field is needed for the line integral to work. Is the line integral supposed to return a scaler, or is it supposed to return a vector? It would make sense to me if it returned a vector, tho i don't remember doing that last year in vector calculus. If it doesn't return a vector, does it return the length of a vector, or how does it transform the vectors from the field into a scaler? Fresheneesz 21:28, 2 August 2006 (UTC)

Ohhh cause its a dot product, I see. I still don't understand it well enough to make the picture any more specific. But you can tell me what I should do (since I have the Imageready file). Fresheneesz 21:28, 2 August 2006 (UTC)

So yea, I like doing pictures to make people understand the concept, but since I don't even understand it very well, I can't make a very good pic. If you want me to make the picture better, i'll make the vector field version better, and I can make a scaler field version. But I need to know what ds is, and exactly what a line integral is. This page needs a better description, and so do I. Lets work to make this page understandable. Fresheneesz 21:33, 2 August 2006 (UTC)

Hi Freshneesz. I unfortunately found you image really misleading - I am a second year degree student, and I used this page, and your animation, to try to answer a set problem; I don't think it is correct though, and had a *lot* of trouble as a result! Please note that I'm not criticising you personally - it's very admirable in my view to contribute an animation. However, I would like to take the current one down, as I worry about it causing people the same problem that it caused me. I will do this if there is no reply or change in the article. I believe ds is the unit normal vector to the path at the point of integration - I hope this is helpful in making a new animation, I am afriad that I don't know how to myself. —Preceding unsigned comment added by 88.106.245.46 (talk) 12:16, 25 December 2007 (UTC)

Line integrals can have vector values. That the particular line integral the previous editor was calculating did not immediately lend itself to assistance from the supplied animation does not make the animation misleading. As such, I don't think that there is that serious a problem with the image to warrant removing it completely, without providing a replacement animation which addresses the issues that the previous editor raised. (Besides, I thought the animation was really good. Removing it wholesale without discussing what has to be done to improve it does not seem like a good policy.) Xantharius (talk) 10:47, 28 December 2007 (UTC)

Hi Xantharius. Whether line integrals can have vector values is a matter of definition, so I guess the issue is what is the most commonly accepted usage by the mathematics community. The article itself doesn't support your statement or the animation, but from your user page it looks like you'd know so I'm guessing that it's the article that isn't as general as it could be. However, all the textbooks I've read (admitting I'm primarily a physicist) define a vector field line integral as it is currently defined in the article; as returning a scalar.

What I'm saying I think is that it would be better to address the issue explicitly and say that vector field path integrals are a broader class of operations, but usually refer to this specific type of operation (involving the dot product). Giving an algebraic definition (and not suggesting it is only part of a wider definition) and then giving a graphical representation of a different object is surely going to confuse. Though someone like yourself would probably see what is going on straight away, I'd imagine that people like you rarely use this article, but rather people like me, who aren't so great at multivariable calculus and are trying to get a clue what is going on. You say that because the article didn't help my problem specifically, that doesn't mean that it's confusing. That's totally true, but I expect many of the people who use it are trying to solve a problem similar to mine.

I have also tried to start a discussion (or at least explain) what could be done to improve the animation; specifically answering Fresheneesz question about what ds is, before removing the animation (I suppose I didn't wait long enough). This post probably looks like a lot of suggestions for other people to do stuff I want changed, and it is really I guess. However, I'm happy to change the article, but I'd just rather someone with the credentials (and knowledge) do it instead, as I don't want any changes just being undone by someone thinking I'm editing for no reason; I genuinely think that this article (specifically the animation) will mislead many of it's viewers (or at least those who are going to use it). —Preceding unsigned comment added by 88.106.173.149 (talk) 03:51, 29 December 2007 (UTC)

I have never heard of line integrals having vector values, though anything is possible. (Well, the complex integral is a kind of line integral, and it can be regarded as vector-valued, since complex numbers can be represented as vectors, but it's not common to speak this way.) But I hope you haven't confused "vector-valued integral" with "vector-valued integrand"... FilipeS (talk) 21:44, 31 December 2007 (UTC)

No, that's not a source of confusion. The integral in the animation is clearly vector valued.

The animation represents the integrand, not the integral. The integral itself along that curve would be a single number. FilipeS (talk) 14:48, 2 January 2008 (UTC)

Yep. And the animation is very pretty. Oleg Alexandrov (talk) 16:23, 2 January 2008 (UTC)

The animation represents an operation, and would be vector valued (if performed as in the animation). It clearly shows the vectors being added without any suggestion of another operation, which would give a vector valued result. I am not confused about the operation or terminology nor am I saying that the animation isn't pretty - people have suggested both that the animation is vector valued and this is fine, and that it is scalar valued (as it should be). To me it clearly shows a vector valued operation, but would be more helpful (and in agreement with the article) if it showed a scalar valued one. Even removing the red pluses would be better as then the animation would just show the integrand. —Preceding unsigned comment added by 88.106.185.22 (talk) 00:02, 3 January 2008 (UTC)

Well, you do have a point. Perhaps the image, nice as it is, would be better suited for one of the articles Vector field, Curve, or Differential geometry of curves. FilipeS (talk) 15:51, 7 January 2008 (UTC)

I'd be happy to change the animation myself, even put in little dot product diagrams for each pair of vectors if someone could recommend a suitable free application. I've looked at a few, but all the animation programs I've seen aren't the right sort of thing. Just something that lets you put in pixels frame by frame would be great. I don't just want to mess up this nice diagram, it's a good sort of thing for the page, but I do think it could be corrected. —Preceding unsigned comment added by 88.106.147.215 (talk) 03:59, 10 January 2008 (UTC)

It would be cool if you could do that. Unfortunately, I am not the right person to advise you regarding the computer application. :-( FilipeS (talk) 20:16, 14 January 2008 (UTC)

Additional examples

I would find it helpful to see examples contrasting when to use $\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.$ versus simply $\int _{a}^{b}f(\mathbf {r} (t))\,dt.$. (Not sure if the latter is technically considered a line integral)

There seems to be confusion over whether the second operation above is a line integral - it seems to be similar to the one in the animation on the article. Could someone who knows for certain please state whether there are several definitions of line integrals on vector fields, or just the 'dot product one'. —Preceding unsigned comment added by 88.106.245.46 (talk) 12:19, 25 December 2007 (UTC)

Look carefully at the two expressions. Clearly, the one on the left is a special case of the one on the right (just define g(r(t)) as the integrand on the left). However, the one on the right is not a subset of the one on the left. Caveat: of course we also need to take into consideration the domains and ranges of f and r. FilipeS (talk) 21:42, 31 December 2007 (UTC)

An additional request: The article includes the complex-plane parameterization example, but what about paths in more than two dimensions? I suggest an example of doing this with a parameterization in, say, $({\hat {i}},{\hat {j}},{\hat {k}})$.

sbump 15:31, 8 August 2006 (UTC)

Font

Shouldn't the font of the title to this article be larger??

I'd change it but I do not know how to.

Rosa Lichtenstein 16:07, 11 September 2007 (UTC)

Do you mean in the formulas? The text font you can change at will in your own browser. FilipeS (talk) 20:17, 14 January 2008 (UTC)

Delete incorrect sentence

The sentence "Line integrals of scalar fields do not depend on the chosen parametrization r." is either totally incorrect or very misleading. Who inserted this sentence? Was it supposed to say "Line integrals of gradients of scalar fields do not depend ..."? If so, then it is in the wrong section and should be moved to the "Path independence" section. You all have one week to complain, then I'm deleting it. 129.78.64.106 (talk) 03:52, 11 June 2008 (UTC)

That sentence is correct. What is misleading about it? Oleg Alexandrov (talk) 05:28, 11 June 2008 (UTC)

Fixed ambiguity RobotMacheen (talk) 04:04, 16 November 2008 (UTC)

Arc length and integrals

Although arc length should not be confused with a line integral, it is part of the definition of any integral. See:

http://knol.google.com/k/john-gabriel/all-integrals-are-line-integrals/nz742dpkhqbi/44#

94.2.151.209 (talk) 11:35, 20 October 2009 (UTC)

I don't agree with the claim in that link that all integrals are line integrals. On the other hand, the caveat "should not be confused with arc length" is probably misguided. If I'm not mistaken, arc length is a special case of the line integral. You get arc length when the integrand in a line integral is the constant (scalar) function equal to unity. FilipeS (talk) 16:33, 2 April 2010 (UTC)

You don't agree with the claim? Well, please tell what you don't agree with. I am not a mathematician, but think the link would add to this article. Do you know what caveat means? I think you may have used the word incorrectly. 114.225.82.14 (talk) 09:43, 18 April 2010 (UTC)

Line integral of a vector field: the definition is wrong

It's about this definition:

$\int _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt.$

While if you just perform formal operations without thinking it may pass, $\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r}$ really makes no sense, because $\mathrm {d} \mathbf {r}$ is not a vector (is it differential of r as a function of t?). The right hand side is correct in $\mathbb {R} ^{n}$.

I think this error needs to be addressed, because it impedes transition of thinking when the reader is introduced to differential forms. I don't know the best way to fix it, some variants I don't like are:

• $\int _{C}\mathbf {F} ^{\flat }$ — I think it's too cryptic, but it's correct.
• $\int _{C}F_{i}\,\mathrm {d} x^{i}$ — It no longer tries to 'dot product' a differential form, but it now instead relies on a cartesian coordinate system with its trivial index lowering.

I think the second variant (maybe with explicit sum) is slightly more acceptable than others, but I'm not sure. Any thoughts? — Kallikanzarid^talk 13:58, 21 January 2011 (UTC)

Now that I think of it, I'm not sure my suggestions are even correct. But I still think the current left hand side of the definition is bad for understanding the subject, and differential forms need to be used a little bit more explicitly — Kallikanzarid^talk 18:39, 21 January 2011 (UTC)

The left hand side is pretty standard notation, I've seen it in numerous places. Conceptually, it makes sense on an informal level; and I don't find it confusing at all. 209.252.235.206 (talk) 10:25, 3 March 2011 (UTC)

"Basically is the area under the constraint..."

Basically is the area under the constraint of the scalar function z=f(x,y), where x=u(t) and y=v(t) is the constraint.

This recent addition to the text is poorly worded, but promising. I think line integrals can be understood as projections. But this needs to be checked, and sourced... FilipeS (talk) 09:29, 14 May 2012 (UTC)

File:Line integral of scalar field.gif to appear as POTD

Hello! This is a note to let the editors of this article know that File:Line integral of scalar field.gif will be appearing as picture of the day on November 12, 2013. You can view and edit the POTD blurb at Template:POTD/2013-11-12. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. Thanks! — Crisco 1492 (talk) 21:47, 23 October 2013 (UTC)

Picture of the day

In mathematics, a line integral is an integral where the function to be integrated, be it a scalar field as here or a vector field, is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).Diagram: Lucas V. Barbosa

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Absolute value of r' for archlength

the formulas currently show |r'| but r is a vector valued function so || is not defined. It should actually be the two norm. — Preceding unsigned comment added by 192.136.116.2 (talk) 22:02, 3 August 2017 (UTC)

Absolute value to denote the length (euclidean norm) of a vector is pretty standard also. I've changed it back to that, especially considering that the rest of the article uses absolute value as well. --Deacon Vorbis (talk) 22:23, 3 August 2017 (UTC)

I'd appreciate if you could give a hint in the definition about the sense of the absoulute value notation use here. Just for all people not common with this but common with the 2-norm notation. Btw, the german article uses the 2-norn notation :-) --Moyalingde (talk) 06:11, 23 November 2017 (UTC)

 Done: Okay, I added a note that hopefully explains it. --Deacon Vorbis (talk) 13:52, 23 November 2017 (UTC)

r or gamma

The uppur part of the lemma uses bold r(t) for the parametrization, the lower uses $\gamma (t)$. I cannot see any reason for this different use. Shouldn't this be unified? I'd prefer to use gamma all over this lemma just to do it the same way as in the german lemma. :-). --Moyalingde (talk) 06:18, 23 November 2017 (UTC)

There are no lemmas in this article, so I'm not sure if we're talking about the same thing exactly, but I noticed that $\gamma (t)$ is used for paths through the complex plane. I think it's good that a separate letter is used here, since it's for a scalar (complex)-valued function, while $\mathbf {r} (t)$ is used for vector-valued (in $\mathbb {R} ^{n}$) paths. --Deacon Vorbis (talk) 13:44, 23 November 2017 (UTC)

Closed loop integral?

I would this article to explain the notation $\oint _{C}\mathbf {F} \cdot d\mathbf {r}$, used to integrate around a closed loop. The-erinaceous-one (talk) 07:38, 8 September 2020 (UTC)