Talk:Curvilinear coordinates

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I might be wrong, but isn't the section on 'Line, surface, and volume integrals' missing all coverage on volume integrals? (I came here because I'm getting the wrong answer when taking volume integrals in spherical coordinates.)

[edit] Is this the same as parametric coords ?

This article is so inaccessible (no pics !) that I can't really understand what it is. Let me describe parametric coords, and you tell me if this is the same thing, please:

• In 1D parametric coords you have a single curve (which can be a line, a 2D curve, a 3D curve, or even a higher dimension curve). The location along the curve is given by the curve length to that point. So, one end of the curve is declared location 0, and the other end is location L, where L is the total curve length. 1D parametric coords are normally used for open curves, but closed curves could work, too, provided a start/end point and direction are selected.

• In 2D parametric coords you have a net of two sets of parallel curves (where each set intersects the other) forming a surface (which can be a planar surface, a 3D surface, or even a higher dimension surface). One corner of the surface has location (0,0) while the other corners have locations (0,V), (U,0), and (U,V), where U and V are the lengths of the curves in the two directions. Again, parametric coords can be used on a surface closed in one or two directions, provided seams can be chosen in each direction.

• 3D or higher dimension parametric coords are also possible, but less frequently used.

So, is this the same thing (in which case this article just needs to be made accessible) or is this something entirely different (in which case it needs it's own article) ? StuRat 01:27, 26 March 2006 (UTC)

It might be better to talk of a parametric surface, this is currently a redirect to Parametric equation but really deserves its own article. For a parametric surface f(x,y) in 'R^3, say f(x,y)=(x,y,x^2-y^2), then the then (x,y,x^2-y^2) are the cartisian coordinates of a point and (x,y) are the parametric coordinates. Parametric coordinates only make sense when you have a parameterisation, a different context to the way curvilinear coordinates are used. --Salix alba (talk) 22:03, 1 April 2006 (UTC)

I see what you mean, but parametric equations aren't just restricted to surfaces. You could also have them for parametric curves or (theoretically) parameterization in any number of dimensions. So, that's why I think a "parametric coordinates" article might make sense, so all cases could be covered at once. Perhaps we still need a "parametric surface" article, as well. StuRat 22:53, 1 April 2006 (UTC)

You're right that you can have quite arbitrary parametrizes spaces. They're called manifolds, and we have an article about them. Parametrized surfaces and curves are studied in detail in elementary calculus, whereas more general parametrized spaces are studied only with the full modern machinery of manifolds. Therefore it doesn't make much sense to have a fully general article called "parametric coordinates". I could get on board an article for parametric surfaces, though. -lethe ^{talk +} 09:50, 2 April 2006 (UTC)

I'm mainly interested in the elementary calculus discussion. How about if the parametric coords article focuses on parametric curves and surfaces, then refers the reader to the manifold article for the more general theoretical discussion ? I picture also creating a redirect from parametric curve (which is now a redirect to differential geometry of curves) and a link from parametric surface (which is a full article now) to parametric coords. StuRat 02:20, 3 April 2006 (UTC)

So what are you saying now? That curves and surfaces don't deserve separate articles? Or would you have them in separate articles, with parametric coords a disambig page between curves, surfaces, and manifolds? I can't think of any material that is appropriate to an article about all three subjects. -lethe ^{talk +} 22:21, 3 April 2006 (UTC)

I'm fine with 3 articles. StuRat 09:17, 8 April 2006 (UTC)

The three articles being manifold, parametric surfaces, and parametric curves? I'm still in the dark about what you would put in a putative fourth article about general parametric spaces. -lethe ^{talk +} 15:42, 8 April 2006 (UTC)

No, the three articles being parametric curve, parametric surface, and parametric coordinates. If you include manifold, that would be four articles total. The parametric coordinates article would include the bullet points from the discussion at the top of this question, with suitable links provided. Essentially, it would be about how one can specify a location on a manifold using parametric coords, with the emphasis being on parametric curves and parametric surfaces. StuRat

If you don't put anything in the article that isn't in curves, surfaces or manifolds, then all you have a disambiguation. The bullets at the top look like disambiguation to me. So far, you haven't mentioned any topic or theorem or construction or notation that would be unique to this hypothetical article, that could justify its existence. Do you have something like that in mind? -lethe ^{talk +} 19:52, 8 April 2006 (UTC)

I've not started Parametric surface, currently focussing on local structure of surfaces in R³. Needs a lot of work but its a start. --Salix alba (talk) 11:41, 3 April 2006 (UTC)

I assume you mean you HAVE started writing Parametric surface ? StuRat 18:15, 3 April 2006 (UTC)

[edit] Expand article

I find this article hard to understand as a person who has not ever heard of curvilinear coordinates - what is it actually about, where is it applied, what do those formulas mean, may it be explained with some pictures? Thanks, --Abdull 18:48, 27 May 2006 (UTC)

Yes, like many of the math/physics pages, this one fails (in my opinion) to give even the vaguest idea of what the word in the title means. A simple diagram could fix this, perhaps showing a 2d system of coordinates as curves equivalent to the 'square grid' that can be drawn with cartesian coordinates. The page is also littered with calculus, where as the fundamental concept of a curvilinear coordinate system doesn't need any calculus to describe or even use. Not that I dislike the additional information, I just question the priorities of the author(s).

AFAIK the main application of such coordinate systems (ignoring polar, cylindrical and other 'special case' curvilinear coordinate systems) is for the mathematical formulation of General Relativity.

Jheriko 16:22, 20 September 2006 (UTC)

[edit] German-speaking Wikipedia

The German article contain a disambiguation page: de:Krummlinige Koordinaten -- Amtiss, SNAFU ? 22:56, 1 November 2006 (UTC)