Talk:Line integral

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1 Requested move
2 Brackets
3 paths and contours
4 vote
5 Equation understanding problems
6 What is it?
7 Additional examples

[edit] 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". --Chan-Ho (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)

[edit] Brackets

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

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

[edit] 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)

[edit] 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)

[edit] 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

$\sum_{1 \le k \le n} f(\gamma(t_k)) ( \gamma(t_k) - \gamma(t_{k-1}) ).$

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)

[edit] 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)

[edit] 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)

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, $(\hat{i}, \hat{j}, \hat{k})$ .