Talk:Integral curve

Definition and figure

There is something not quite right about the definition given at the outset and the figure to the right at the start of the article suggests what I am concerned about.

I might be missing something simple or essential here, but I am bothered by the way this article chooses to actually define what in integral curve is at the outset.

The article makes the claim that an integral curve is by definition a "parametric curve." Indeed, once past the opening remarks, all of the technical definitions and elucidations are in terms of parametric representations.

I would be the first to agree that a parametric curve is sometimes a handy representation of any curve, but as the opening plot demonstrates, you can label and plot integral curves for illustrative purposes without ever identifying a parameterization or even a definite parameter.

In dynamics problems, the DE's necessarily have a time "parameter" and aren't flows essentially integral curves parameterized with time? I understand that in principle there exists some parameterization that would generate the three specific integral curves for each of three initial conditions as shown imposed on the vector field background. But if you can find a way to plot out integral curves in some cases without ever selecting or inventing a suitable parameter, what does that say about how essential parametric representations of integrals curves is the the definition of what an integral curve is?

I thus question whether defining an integral curve as first and foremost a "parametric curve", albeit one that maps the solutions of a particular differential equation, is helpful or definitive. All you can really say is that if you have a set of DE's already in a form that you can recast into parametric form, then you can plot the integral curves in a straightforward way. I think, however, that in real life, finding an explicit parametric representation of an arbitrary non-trivial DE is not guaranteed to always be easy unless the parameter is already imbedded into the DE's to start with. --scanyon (talk) 05:38, 30 March 2013 (UTC)

"Solution curve"

Solution_curve redirect to Integral_curve.

But "solution curve" is not mentioned in the text of the article. Is solution curve the same as integral curve?

In any case we need to define what "solution curve" means in the text of the article.

--VictorPorton (talk) 19:35, 4 August 2016 (UTC)

Example, please!!!

Most of us on Wikipedia do not possess an abstract mind! — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 15:15, 30 December 2019 (UTC)