Talk:Curvilinear coordinates/Archive 1

Archive 1

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)

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)

German-speaking Wikipedia

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

Suggested Modification

Do you think that the first sentence of the General curvilinear coordinates section...
"In Cartesian coordinates, the position of a point P(x,y,z) is determined by the intersection of three mutually perpendicular planes, x = const, y = const, z = const."
...would be better expressed as...
"In Cartesian coordinates, the position of a point P(x₀,y₀,z₀) is determined by the intersection of three mutually perpendicular planes, x = x₀, y = y₀, z = z₀."
This seems like a more correct/clear wording to me, although I'm not sure if it's what was intended (not quite confident enough to change it myself, without checking with someone else). Note: I may not come back here, so if you think this is a good improvement please go ahead and change it yourself.DonkeyKong the mathematician (in training) 10:32, 6 May 2007 (UTC)

Upon further reflection, (a,b,c) might be better than (x₀,y₀,z₀). DonkeyKong the mathematician (in training) 10:35, 6 May 2007 (UTC)

Since this is a discussion of general coordinate system, I have abstained from using indices, or different letters, other than coordinates, so as not to confuse the reader what the actual coordinates are. The main point is that (x,y,z) are the well-known Cartesian coordinates while (q₁,q₂,q₃) are the curvilinear coordinates to be defined here. I even thought to avoid the indexes on the curvilinear coordinates, marking them, e.g. (u,v,w) but didn't do it because I felt it could take out some of their generality. This is, however, still a very rough variant of changes that I have planned and any suggestions are welcome.--Lantonov 05:48, 7 May 2007 (UTC)

-

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.)

Yes, the article misses volume integrals and a lot more. Slowly adding info, though (sigh). --Lantonov 08:21, 11 June 2007 (UTC)

Figure 2 confusion

Figure 2 appears to use left-handed cartesian coordinates and swaps the usual zenith (φ) and azimuth (θ) notation (cf. Spherical coordinate system). The coordinate transformations described in the text do however, use the standard notation. I recommend exchanging the current figure with Image:Spherical Coordinates.png. Mtb80 16:18, 7 June 2007 (UTC)

In most texts (cf. eg Korn & Korn, Arfken, Borisenko & Tarapov, Margenau & Murphy), φ is the angle in the horizontal plane, and θ is the angle in the vertical plane, as in Fig. 2. The figure in Spherical coordinate system goes against this convention and swaps φ and θ angles. However, Mtb80 is right that in math texts (as opposed to physics texts) the right-handed Cartesian coordinates are used. Accordingly, I swapped the x and y axes in Fig. 2. I do not recommend exchanging the current Fig. 2 with Image:Spherical Coordinates.png because the latter misses all the elements described in the text, namely the spherical coordinate lines, surfaces and axes. Lantonov 06:53, 8 June 2007 (UTC)

The formulas immediately to the left of Fig. 2 for the direct and indirect transformations nevertheless use the opposite convention for φ and θ. The lack of correspondence between these formulas and the definitions of φ and θ immediately preceding them (which correspond with the current Fig. 2) are the source of the confusion. If the international convention cited above is to be used - and I agree it should - I recommend exchanging φ and θ in the trasformation formulas. Mtb80 14:30, 19 June 2007 (UTC)

Mtb80, you are right, my error. On your suggestion, I exchanged φ and θ after consulting with Arfken, Mathematical Methods for Physicists, 2001, p. 121 and p. 123. Now the transformation formulas and the figure correspond exactly to this book. Lantonov 16:46, 19 June 2007 (UTC)

I also checked Korn & Korn, Mathematical Handbook for Scientists and Engineers. The equations and picture are the same as in Arfken and now as in this article. Thank you for pointing this out. Lantonov 05:12, 20 June 2007 (UTC)

Clearly not for Mathematicians

I am a mathematician and this page makes absolutely no sense. I appreciate the applied approach, but perhaps a second section to this article is needed that summarizes this topic from a more abstract mathematical point of view. The notion of tangent bundles seem to be crucial to defining coordinate systems, and all I see are the words "moving basis". —Preceding unsigned comment added by 130.58.245.171 (talk) 07:00, 8 May 2008 (UTC)

Of course, this article is not for mathematicians. It is for the general public and that's why it is in an encyclopaedia which is read by anybody. If you want very abstract and rigorous exposition, I suggest you read specialized mathematical literature. About things making no sense, you have to be more specific in order to correct what is wrong or inaccurate. As a mathematician, your effort in improving the article will be much appreciated. I didn't find "moving basis" here. Maybe, you mean "local basis". This is a term used in "serious" books. It will be nice if you introduce the notion of tangent bundles. There is scarcity of references here, especially intext, which is a seriuos defect. I have consulted well known monographs for text that I wrote, and one day I will find the time to reference them. --Lantonov (talk) 05:36, 9 May 2008 (UTC)

Invertible or locally invertible?

An invertible transformation means that there is a one-to-one correspondence (bijection) between the curvilinear and Cartesian coordinates. A locally invertible transformation means that there is such a relationship for an infinitesimal change in the curvilinear and Cartesian coordinates. It would seem that this article confuses this at several places, noticeably in the opening paragraph, where the third sentence does not follow from second sentence but instead requires a globally invertible transformation. In particular, polar coordinates are locally invertible, but are only globally invertable if the radial coordinate is restricted to be positive, and the angluar coordinate is restricted to the domain $[0,2\pi )$. Such a restriction has the unfortunate side effect of excluding the origin. Another problem is demonstrated in the example figure, where the $g_{1}$ coordinate lines appear to cross. If we require global invertibility, this is not allowed. In general, I think the transformation from curvilinear to Cartesian coordinates only needs to be onto (surjective) and smooth. 128.146.35.111 (talk) 00:32, 6 February 2012 (UTC)

In my opinion your comment should appear as a note in the article and the third sentence should add a qualifier that says that the map is only locally a diffeomorphism. Bbanerje (talk) 22:39, 7 February 2012 (UTC)

change of notation

Another problem which gives a blaring, detialed appearance is the use of the absurd letter ξ for the coordinates! Why not just use the standard q?? I changed this as its far more typographically conveient, visually cleaner, easier to handwrite (if someone takes notes from this article... they will have to struggle with all the squiggles and coils in ξ... its not easy), and it also matches with the lead description (and has a nice link with generalized coordinates, even though they are not the same they still have the "general" idea).

Also changed the curvillinear basis vectors g to b for no confusion with the metric, notationally cleaner, b is the first letter of basis, and since b and q are π rad rotations of each other about any point in the plane of those letters! So b is a better (!) choice than g.

In doing so - I also tweaked the images, and made the lines smoother in the first one. Hope no one minds...F = q(E+v×B) ⇄ ∑_ic_i 09:42, 28 April 2012 (UTC)

Removal of content

I can't take it any longer. Some repeated and/or unnecessary sections will simply be removed for now, then the article will be shortened as much as possible. Some sections are simply examples which are pointless repetitions of what comes before them and/or simply special cases which could be worked out using the endless general rules in the article anyway. Hence I will be deleting the following:

1. Relations between curvilinear and Cartesian basis vectors - this can be worked out anyway
2. Vector products - again??
3. Example: Cylindrical polar coordinates - simply unreadable

On the other hand, the following will definitely be kept, but rewritten:

1. Curvilinear local basis
2. switch the order of Vector and tensor calculus in three-dimensional curvilinear coordinates and Orthogonal curvilinear coordinates, and formidably shorten the stuff on the vector calculus (line/surface integrals, nabla operators + generalization to tensorial objects, worth mentioning but should only summarized at a maximum, they'll be tabulated/listed neatly)
3. move the template:Orthogonal coordinate systems to the bottom, why stuck in the middle??
4. Other bits will be merged.

After this a decision can be made on creating new articles if needed. F = q(E+v×B) ⇄ ∑_ic_i 15:32, 26 April 2012 (UTC)

(Talking to myself again...) Never mind... Not sure if there is any need to move anything now that the article has lost ~31kB of repetition/emphasis on inessential details... Most of the tensor mass of detail has been eliminated since it’s not that much help, it’s just too dense to have so many little intricate derivations for an encyclopaedia article. In any case, at least there seems to be some orderly structure to the article rather than chaos. All that’s really needed is clean up. Yes I will add references just in case someone comes along, but after clean up, else referenced bits may just be thrown hence wasting time... F = q(E+v×B) ⇄ ∑_ic_i 22:41, 28 April 2012 (UTC)

Most of the stuff that's been deleted is essential for large deformation continuum mechanics. That information is usually dispersed over the length of a textbook. I suggest that you revert those changes or at least put them in an article called "Curvilinear coordinates (continuum mechanics)". Orthogonal curvilinear coordinates are not very useful in continuum mechanics except for some special cases. Bbanerje (talk) 10:06, 30 April 2012 (UTC)

It was a mash of unreadable tensor calculus which was not essential for understanding what curvilinear coordinates are - the subject of this article, not its applications (to Continuum mecdhanics, as you say). Not everything should be obfuscated with tensors, as its easily possible to talk about curvillinear coordinates using vector calculus, with some mention of mathematical results and applications. This article is probably still too big as it is. Also orthogonal coordinates are the easiest to learn and use as an introduction, hence my own emphasis.

I did propose to place the tensor stuff into another article Tensors in curvilinear coordinates but no-one responded. If you really want that material back I will dig it back up from the edit history, add it to the proposed article, clean up as much as I can, and it's all yours from there. Agreed? Thanks for becoming involved - much apprciated. F = q(E+v×B) ⇄ ∑_ic_i 11:13, 30 April 2012 (UTC)

It has been created, though not sure how you would like the content to be organized, what more to add/subtract etc. Please advise/see to it. F = q(E+v×B) ⇄ ∑_ic_i 12:15, 30 April 2012 (UTC)

problems with the article

Here are a few problems:

1. Its far too long and dense (~87,319 bytes). There is just so much text and equations blaring from the screen that its unreadable. (no clue how a typical reader can follow all this... becuase its so long and wordy...)
2. the lead will loose so may readers since its a mash of both vector and tensor formalism.
3. The lead has an unnecerssary amount of detial for the lead (each section for that matter).
4. Some operations seem to be repeated (the cross product is)

It would be much more instructive to start with this approch:

1. Lead on coordinate curves, surfaces and transformations
2. State + link the Jacobian - while important and relevent, its the subject of another article.
3. Begin with the vector in orthogonal coords:

   $\mathbf {r} =\mathbf {r} (u_{1},u_{2},u_{3})=u_{1}\mathbf {e} _{1}+u_{2}\mathbf {e} _{2}+u_{3}\mathbf {e} _{3}$

   (although this article uses $\xi$ and q, it seems u is standard).
The differential is

$d\mathbf {r} ={\dfrac {\partial \mathbf {r} }{\partial u_{1}}}du_{1}+{\dfrac {\partial \mathbf {r} }{\partial u_{2}}}du_{2}+{\dfrac {\partial \mathbf {r} }{\partial u_{3}}}du_{3}=h_{1}du_{1}\mathbf {e} _{1}+h_{2}du_{2}\mathbf {e} _{2}+h_{3}du_{3}\mathbf {e} _{3}$

where

$h_{1}=\left|{\frac {\partial \mathbf {r} }{\partial u_{1}}}\right|,h_{2}=\left|{\frac {\partial \mathbf {r} }{\partial u_{2}}}\right|,h_{3}=\left|{\frac {\partial \mathbf {r} }{\partial u_{3}}}\right|$

are the scale factors. State also their relation to tangent vetors along the coord curves:

$\mathbf {e} _{1}={\dfrac {\frac {\partial \mathbf {r} }{\partial u_{1}}}{\left|{\frac {\partial \mathbf {r} }{\partial u_{1}}}\right|}}={\frac {1}{h_{1}}}{\frac {\partial \mathbf {r} }{\partial u_{1}}}\cdots$

4. then state the differential length element, area element and volume element and explain a few properties.
5. Then, given the position vector in curvillinear coords, state the dot and cross product, give the forumula for the gradient, hence div and curl.
6. Then some examples.
7. Then and only then generalize to tensors...

I'm going to re-write. Its simply unreadable... F = q(E+v×B) ⇄ ∑_ic_i 08:05, 12 April 2012 (UTC)

Forgot to mention - use show/hide boxes for encapsulating content when:

1. the article becomes too long,
2. content would not be essential, but useful/helpful and relevant for interested readers,
3. it's not easy to find somewhere else better,

perhaps on some of the tensor formalism later, but not for now. F = q(E+v×B) ⇄ ∑_ic_i 08:16, 12 April 2012 (UTC)

Although I appreciate the effort of whoever drew File:Spherical coordinate elements.svg (currently in the article), these by user:WillowW:

may be clearer and less cluttered for illustrating coordinate curves and surfaces. I'll not change that for now, just a suggestion. If no-one objects in the future, I'll just add replace them myself. F = q(E+v×B) ⇄ ∑_ic_i 10:31, 12 April 2012 (UTC)

On second thought its difficult to decide exactly what to throw away, since much of the material is nicely explained, though could still do with shortening. Even trimming it down would not shorten the that article much though... Perhaps leave all the vector algebra and coordinate descriptions here... and cast a new article Tensors in curvilinear coordinates for all the tensor formalism?..... F = q(E+v×B) ⇄ ∑_ic_i 16:00, 12 April 2012 (UTC)

The above crossed out is now irrelevent. The proposed article is now new. F = q(E+v×B) ⇄ ∑_ic_i 17:39, 1 May 2012 (UTC)

merge with orthogonal coordinates

There is much overlap between the two articles. Although curvilinear coordinates (g-tensor not necessarily diagonal) are somewhat more general than orthogonal coordinates (diagonal g-tensor), the examples and the operators (div, curl, Laplace) coincide in the two articles. Basically both articles are on classical tensor analysis.--P.wormer 16:50, 22 December 2006 (UTC)

Orthogonal and curvilinear coordinates are two very different things. The coincidence of formulas is only apparent. Personally, I too think that this article misses a main thrust and is overburdened with advanced formulas lacking explanations and pictures. Most of the confusion springs out from this. One wishes to see the difference between linear and curvilinear vectors, tensors, curl etc. explained in a manner that is easier to understand than most standard (advanced) texts. --Lantonov 08:27, 26 April 2007 (UTC)

I think that this is an important issues for this page. For example the expressions given in the Curvilinear_coordinates#Differentiation section are only valid for the orthogonal case, not in the general case. This is clearly misleading as someone could easily think that these expressions are the general ones. Even more so given that the same expressions also appear on the page on orthogonal coordinates. At least this section should be rewritten to make this clear. --Popinet 06:51, 14 May 2012 (UTC) — Preceding unsigned comment added by 219.89.38.133 (talk)

Question in section 1.1

In section 1.1 Coordinates, basis, and vectors, it says that in curvilinear coordinates, any vector can be represented as

$\mathbf {r} =h_{1}q_{1}\mathbf {b} _{1}+h_{2}q_{2}\mathbf {b} _{2}+h_{3}q_{3}\mathbf {b} _{3}$

I tested this out with spherical coordinates. For spherical coordinates, h1=1, h2=r, h3=rsinθ. According to the above equation, this should mean that

$\mathbf {r} =r\mathbf {e} _{r}+r\theta \mathbf {e} _{\theta }+r\sin \theta \phi \mathbf {e} _{\phi }$

but that obviously isn't the case because $\mathbf {r} =r\mathbf {e} _{r}$

Have I done something wrong? Or is the equation wrong? I know that for infinitesimal r then it's true, but I don't think it's true for non-infinitesimal r thedoctar (talk) 02:27, 10 January 2013 (UTC)

You are absolutely correct - this is a major error on the page. I suggest the following replacement for this part; I'll give this a few days and then (try to remember to) replace it if there are no objections:

The relation between the coordinates is given by the invertible transformations:

$x=x(q_{1},q_{2},q_{3}),\,y=y(q_{1},q_{2},q_{3}),\,z=z(q_{1},q_{2},q_{3})$

$q_{1}=q_{1}(x,y,z),\,q_{2}=q_{2}(x,y,z),\,q_{3}=q_{3}(x,y,z)$

Any point can be written as a position vector r in Cartesian coordinates:

$\mathbf {r} =x\mathbf {e} _{x}+y\mathbf {e} _{y}+z\mathbf {e} _{z}$

where x, y, z are the coordinates of the position vector with respect to the standard basis vectors e_x, e_y, e_z.

However, in a general curvilinear system, there may well not be any natural global basis vectors. Instead, we note that in the Cartesian system, we have the property that

$\mathbf {e} _{x}={\dfrac {\partial \mathbf {r} }{\partial x}};\;\mathbf {e} _{y}={\dfrac {\partial \mathbf {r} }{\partial y}};\;\mathbf {e} _{z}={\dfrac {\partial \mathbf {r} }{\partial z}}.$

We can apply the same idea to the curvilinear system to determine a system of basis vectors at P. We define

$\mathbf {h} _{1}={\dfrac {\partial \mathbf {r} }{\partial q_{1}}};\;\mathbf {h} _{2}={\dfrac {\partial \mathbf {r} }{\partial q_{2}}};\;\mathbf {h} _{3}={\dfrac {\partial \mathbf {r} }{\partial q_{3}}}.$

These may not have unit length, so we define the Lamé coefficients (after Gabriel Lamé) by

$h_{1}=|\mathbf {h} _{1}|;\;h_{2}=|\mathbf {h} _{2}|;\;h_{3}=|\mathbf {h} _{3}|$

and the curvilinear basis vectors by

$\mathbf {b} _{1}={\dfrac {\mathbf {h} _{1}}{h_{1}}};\;:\mathbf {b} _{2}={\dfrac {\mathbf {h} _{2}}{h_{2}}};\;:\mathbf {b} _{3}={\dfrac {\mathbf {h} _{3}}{h_{3}}}.$

It is important to note that these basis vectors may well depend upon the position of P; it is therefore necessary that they are not assumed to be constant over a region. (They technically form a basis for the tangent bundle of $\mathbb {R} ^{3}$ at P, and so are local to P.)

Julian Gilbey (talk) 23:58, 2 February 2013 (UTC)

Looks fine to me. You haven't replaced it yet, so I'm going to presume that you forgot and replace it myself.thedoctar (talk) 09:31, 10 February 2013 (UTC)

Well, with further thought, one only defines the Lamé coefficients for orthogonal curvilinear coordinates, so I guess that should go there too.

Julian Gilbey (talk) 17:00, 10 February 2013 (UTC)

Well, I'll trust you on that! I don't know non-orthogonal coordinates. Don't forget to fix the inaccuracies you mentioned below; they're mostly about tensors so I can't help you out there either.

thedoctar (talk) 09:05, 11 February 2013 (UTC)

Further issues with the page

I have further problems with this page. Some of them, I have ideas about, others I am less sure. I'd love an opinion from an expert in this area before making any changes.

• Section 1.4, Covariant basis, has a number of issues.
  • Section 1.4.1, Constructing a covariant basis in one dimension, seems to assume that the $q_{1}$ axis is nicely lined up with the x-axis so that the triangle PAB is right-angled; there seems to be no reason whatsoever to assume that this is the case. Furthermore, as earlier it was stated that $\mathbf {b} _{1}$ is a unit vector, this is even stranger! I have no idea what the author of this section intended - there may be some construction along these lines which is well-known.
  • Section 1.4.2, Constructing a covariant basis in three dimensions, seems to be full of incorrect equations - the $\mathbf {b} _{i}$ should either be replaced by $\mathbf {h} _{i}$ in the first set of equations or the right hand sides need dividing by $h_{i}$ to normalise the vectors. For the inverse transformation, each term needs dividing by one of the $h_{i}$, I believe.
  • I am not clear what the purpose of the Jacobians in the next subsection is - they are most usefully used when changing the coordinate system for an integration, and then this is just the standard Jacobian for substitutions, so really belongs on a page about integration by substitution.

• Section 2, Generalization to n dimensions, seems somewhat out of place: most of the page is talking about three-dimensional calculus. Perhaps this could be moved to later on? And if it belongs on this page (which it might), it should have some sort of description of which of the 3D results generalise (and how) to n dimensions.

• Section 3.2, Vector and tensor algebra in three-dimensional curvilinear coordinates, seems to be a bit of disaster.
  • In section 3.2.1, Tensors in curvilinear coordinates, second-order (or rank two) tensors are defined and related using the various metric tensors, even though these are not defined until the next subsection.
  • In section 3.2.2, The metric tensor in orthogonal curvilinear coordinates, the vector $\mathbf {b} _{i}$ is used, even though its definition differs from that earlier on the page (where it was a normalised vector); surely the notation $\mathbf {g} _{i}$ would be better for this (it may also be more standard)?
  • Also, the title of this subsection is strange: why restrict it to the orthogonal case? This is the definition in general.
  • Finally for this subsection, the index-raising formula is given, but not the index-lowering one.
  • In section 3.2.3, Relation to Lamé coefficients, the notation is somewhat confused. If we define $h_{k}^{i}$ as in section 1.2.1 Differential Elements (and not $h_{ij}$ as used here), then the second equation is correct. I also believe (though I may be wrong) that Lamé coefficients are only used for the orthogonal case, in which case the presence of $h_{i}$ in the first equation is misleading; it can be (re)introduced in the final line in this subsection.
  • In section 3.2.4, Example: Polar coordinates, the example is two-dimensional where all of the theory has been three-dimensional (though this may be considered a minor issue).

Julian Gilbey (talk) 01:36, 3 February 2013 (UTC)

This article has seen a steady increase in entropy over time and needs detailed rechecking and verification. Readers and editors appear to prefer orthonormal bases and that preference is also reflected in the article. Please go ahead and fix if you have the time and inclination. Bbanerje (talk) 21:58, 3 February 2013 (UTC)

Orthonormal bases work fine for orthogonal curvilinear coordinates, but not for general curvilinear coordinates, where the coordinate curves are not, in general, orthogonal. Julian Gilbey (talk) 00:22, 4 February 2013 (UTC)

Stumbled here while editing potential gradient. Although not an expert mathematician, I know orthogonal coordinates and to a lesser extent general curvilinear coordinates, and would be willing to help make this important article better.

On a minor point, are there problems with the diagram I drew/added? There shouldn't be, just asking for other people's opinions for improvement if needed. Best regards, M∧Ŝc²ħεИ_τlk 16:58, 5 March 2013 (UTC)

I've been meaning for ages to rewrite this article. Unless someone beats me to it - in the next few days I intend to either

• blank the page (apart from the references and diagrams) and start over (anything useful will be reinstated after, and anyone is of course welcome to add useful back stuff from the history), or
• revert to a much older version of the page if that turns out to be better,

given the numerous issues with the page.

• Correct the equations.
• Completely re-organize content in a hopefully more logical order: mappings between Cartesian and curvilinear coordinates, definitions and geometric interpretations of co- and contra- variant bases and components, motivate the need for the metric tensor and Jacobian matrices and determinant, relation between Jacobian det and metric det, then the scalar triple product and permutation symbol in curvilinear coordinates, then go onto dot and cross products, then vector calculus of scalar and vector fields (grad, div, curl, Laplacian operators, line, surface, volume integrals and associated line, surface, volume elements), then tensor calculus (covariant and absolute derivatives). If this ordering of content is unsatisfactory - does anyone have better ideas?
• Clarify and simplify the notation.
• De-emphasize orthogonal coordinates and provide transitions from Cartesian to curvilinear coordinates,
• Write most of the article in 3d for concreteness, simplicity, and readily applicable content (generalizations to arbitrary dimensions left to the end).

M∧Ŝc²ħεИ_τlk 23:42, 18 February 2014 (UTC)

Statement in introduction is wrong

If I am not mistaken, the statements

In two dimensional Cartesian coordinates, we can represent a point in space by the coordinates ($x_{1},x_{2}$) and in vector form as $\mathbf {x} =x_{1}~\mathbf {e} _{1}+x_{2}~\mathbf {e} _{2}$ where $\mathbf {e} _{1},\mathbf {e} _{2}$ are basis vectors. We can describe the same point in curvilinear coordinates in a similar manner, except that the coordinates are now ($\xi ^{1},\xi ^{2}$) and the position vector is $\mathbf {x} =\xi ^{1}~\mathbf {g} _{1}+\xi ^{2}~\mathbf {g} _{2}$. The quantities $\xi ^{i}$ and $x_{i}$ are related by the curvilinear transformation $\xi ^{i}=\varphi _{i}(x_{1},x_{2})$. The basis vectors $\mathbf {g} _{i}$ and $\mathbf {e} _{i}$ are related by

$\mathbf {g} _{i}={\cfrac {\partial x_{1}}{\partial \xi ^{i}}}\mathbf {e} _{1}+{\cfrac {\partial x_{2}}{\partial \xi ^{i}}}:\mathbf {e} _{2}$

are wrong. Position vectors do not behave like tangent vectors. --Trigamma (talk) 17:00, 14 February 2012 (UTC)

Sorry for my english! i think that Jacobian is wrong! The indices of elements of J matrix /Jacobian of the transformation/ there are wrong. I think the correct form

$\mathbf {J} ={\begin{bmatrix}{\cfrac {\partial x_{1}}{\partial q^{1}}}&{\cfrac {\partial x_{2}}{\partial q^{2}}}&{\cfrac {\partial x_{3}}{\partial q^{3}}}\\{\cfrac {\partial x_{1}}{\partial q^{2}}}&{\cfrac {\partial x_{2}}{\partial q^{2}}}&{\cfrac {\partial x_{3}}{\partial q^{2}}}\\{\cfrac {\partial x_{1}}{\partial q^{3}}}&{\cfrac {\partial x_{2}}{\partial q^{3}}}&{\cfrac {\partial x_{3}}{\partial q^{3}}}\\\end{bmatrix}},\quad$

Have a nice day Slamo — Preceding unsigned comment added by 91.144.81.35 (talk) 12:21, 14 October 2014 (UTC)

Non-curvilinear coordinates?

The present definition seems pretty broad -- includes Cartesian, angular (polar, spherical), cylindrical, all of orthogonal and non-orthogonal (skew). What then would be left out -- homogeneous coordinates only? Worth discussing in the article? Thanks. Fgnievinski (talk) 22:55, 11 November 2014 (UTC)

It might be worth mentioning non-curvilinear coordinate systems. Curvilinear coords typically demand some differentiability conditions so you can do calculus on them. So curvilinear excludes non-smooth coordinates, like position along a fractal or random walk. When the Jacobian becomes degenerate at given points (what's the longitude at the North Pole?), invertibility fails and and at these singular points one could say that curvilinearity breaks down. The last example I can think of is non-metric spaces where dimensions are incomparable. An example would be pressure-volume diagrams in thermodynamics. While one can consider surfaces of constant pressure or volume as defining coordinates, different units mean there is no rotating or transforming of these coordinates that in any way mixes P and V. --Mark viking (talk) 00:02, 12 November 2014 (UTC)

Normalization of b_i and bⁱ basis

The basis vector $\mathbf {b} _{i}={\dfrac {\partial \mathbf {r} }{\partial q_{i}}}$ and $\mathbf {b} ^{i}=\nabla q_{i}$ cannot be normalized if one wants to keep the very important dot product rule $\mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}$.

Indeed, two vectors of unit length and whose dot product is equal to one have necessarily the same direction (cos θ = 1), meaning that b_i and bⁱ are colinear, which trivially is not the case for all curvilinear coordinate systems.

Also, the previous version of the article (corrected meanwhile) assumed that $\left|{\dfrac {\partial \mathbf {r} }{\partial q_{i}}}\right|={\dfrac {1}{\left|\nabla q_{i}\right|}}$, which is not correct.

The normalized/un-normalized notation used in this article seems inconsistent. The first two sections of this article define $\mathbf {h} _{i}={\dfrac {\partial \mathbf {r} }{\partial q_{i}}}$ and $\mathbf {b} _{i}={\dfrac {\mathbf {h} _{i}}{h_{i}}}$ implying that b is the normalized version of h. Yet in the very next section, the definition $\mathbf {b} _{i}={\dfrac {\partial \mathbf {r} }{\partial q_{i}}}$ contradicts the above definition. I think ${\hat {\mathbf {b} }}$ or ${\hat {\mathbf {h} }}$ should be used to better differentiate normalized vectors. - 108.5.142.2 (talk) 19:35, 16 December 2020 (UTC)

Misuse of the Lame coefficients

In "3. General curvilinear coordinates in 3D" the Lame coefficients are "defined" by h_i h_j = g_ij which has generally no solution because there are six independant equations (for g_11, g_22, g_33, g_12, g_13 and g_23) and only three unknowns (h_1, h_2, h_3).

In orthogonal curvilinear coordinates (g_12=g_23=g_13=0), the first three equations define h1, h2, h3 but the last three are not verified (h1h2 differs from 0!) so this "definition" is false.

I think that lame coefficients should not appear in part 3 where non orthogonal coefficients are included, and I modified the comment after their (correct) definition in part 1, that was intended to generalize them to non-orthogonal systems. They are used in part 4 (vector and tensor calculus in 3d curvilinear coordinates), which should be restrained to orthogonal coordinates if such formulas are used. In this part 4, some "lamé coefficients" h_ij are used, maybe in place of the covariant components g_ij of the metric tensor...

Part 4 has certainly to be corrected...Yves.Delannoy (talk) 16:20, 19 February 2016 (UTC)

I think $h_{i}h_{j}=g_{ij}$ was meant to be $\mathbf {h} _{i}\cdot \mathbf {h} _{j}=g_{ij}$. This definition would be consistent with the definitions given in section one. Although $\mathbf {h} _{i}\cdot \mathbf {h} _{j}=g_{ij}=\mathbf {b} _{i}\cdot \mathbf {b} _{j}$ is redundant. 108.5.142.2 (talk) 19:44, 16 December 2020 (UTC)

If I recall correctly, the section originally had $\mathbf {g} _{i}$ etc. as the basis vectors with a small description of the relation between the $\mathbf {g}$ and other notations used in the original (orthonormal coordinates only) version of the article. The article has been modified significantly since then without much consideration for consistency of notation and needs to be gone thru with a fine-toothed comb to make things consistent again. Bbanerje (talk) 20:50, 16 December 2020 (UTC)