Talk:Curvilinear coordinates

Scale Factor

Somebody please put in scale factor for orthogonal curvilinear coordinate case to make it easier to recognize.

• cartisian to u, v, w coordinate system

x=x(u,v,w) , y=y(u,v,w), z=z(u,v,w) |hu|=sqrt[(dx/du)^2+(dy/du)^2+(dz/du)^2] & so on.....

Covariant Basis Section and Later Needs Improvement

The index structure of the x coordinates and q coordinates are not the same! This is incorrect and needs to be fixed. This begins in the Covariant basis section and propagates through the rest of the article.

Figure 3 incorrectly shows dq as the hypotenuse of the infinitesimal triangle. The coordinate q is a function of x and as such the coordinate differential dq is the height of the infinitesimal triangle.

Covariant and contravariant confusion

About the § "Covariant and contravariant bases"

The position of the indices concerns only the coordinates and not the basis vectors (in fact, because of Einstein's convention, it is even the opposite).

For the vectors we have the définitions "vectors are contravariant" and "covectors are covariant". This is in accordance with the fact that a vector is a tensor of type (1,0) and a covector is a tensor of type (0,1).

Thus the basis $(b_{1}\cdots b_{n})$ (which gives the contravariant coordinates) is a contravariant basis.

In addition, in the main article reported in the article, it is said that the gradient is a covector and that the covectors are covariant. Since $b^{i}$ is a gradient, it is therefore a covariant vector and the basis $(b^{1}\cdots b^{n})$ (which gives the covariant coordinates) is a covariant basis.--KharanteDeux (talk) 15:23, 2 August 2021 (UTC)

This creates contradictions in other elements of the same wiki page.

Further down in the main article it says:

"A vector can be specified with covariant coordinates (lowered indices, written v_k) or contravariant coordinates (raised indices, written v^k). From the above vector sums, it can be seen that contravariant coordinates are associated with covariant basis vectors, and covariant coordinates are associated with contravariant basis vectors."

Referring to:

"A vector v can be specified in terms of either basis, i.e.,

$\mathbf {v} =v^{1}\mathbf {b} _{1}+v^{2}\mathbf {b} _{2}+v^{3}\mathbf {b} _{3}=v_{1}\mathbf {b} ^{1}+v_{2}\mathbf {b} ^{2}+v_{3}\mathbf {b} ^{3}$"

Applying the designation that you suggest leads to contravariant components being paired with contravariant basis vectors. From how I understand it, this is incorrect.

While I agree that vectors are called co- or contravariant vectors with respect to their coordinates, it is common to label basis vectors being co- and contravariable as well.

I suggest further discussions but I think its necessary to undo the change by @KharanteDeux on 15:49, 23 October 2021‎. Fepau (talk) 12:18, 23 November 2022 (UTC)

Milena 191.156.246.123 (talk) 03:49, 11 March 2023 (UTC)

Intro confusing

For someone who doesn't already know what curvilinear coordinates are, the intro is confusing, especially the second sentence:

These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point.

1. The article should mention at the very beginning that each point will have its own coordinate system, since this is what curvilinear coordinates are about in the first place (i. e. curvilinear coordinate systems are local coordinate systems).
2. It is unclear how the mentioned transformation can lead to the curvilinear coordinates (it does by differentiation).
3. If the local coordinate systems for all points are defined by a single function in this way, local invertibility is not enough; it needs to be a differentiable transformation and the total derivative needs to be invertible at every point.
4. It sounds like the local coordinate system at a point P was to be used to get a coordinate representation of the point P itself, while in most application the curvilinear coordinates are only useful if the coordinate system at the point P is used to represent a vector different from P, for example the value f(P) of some function f.

I will slightly rewrite that part of intro unless there are objections to any of these points.

Ninjamin (talk) 18:17, 29 January 2023 (UTC)

confusion of \vec h_i and \vec b_i

It looks like someone tried to change from the more tradition \vec b_i=\partial \vec r/\partial q^i to a seemingly more consistent \vec h_i = \partial \vec r/\partial q^i, so that its magnitude is h_i . However the fact that b^i is also used (for the reciprocal basis) means that you would also need an h^i=1/h_i, which is not consistent with the literature. Thus I feel this new notation is confusing. Also, this notation of \vec h_i replaced \vec b_i in some, but not all sections of the article, further compounding the confusion, see, for example: Covariant and contravariant bases. In what I have seen, it is It looks like someone tried to change from the more tradition \vec b_i=\partial \vec r/\partial q^i to a seemingly more consistent \vec h_i = \partial \vec r/\partial q^i, so that its magnitude is h_i . However the fact that b^i is also used (for the reciprocal basis) means that you would also need an h^i=1/h_i, which is not consistent with the literature. Thus I feel this new notation is confusing. Also, this notation of \vec h_i replaced \vec b_i in some, but not all sections of the article, further compounding the confusion, see, for example: Covariant and contravariant bases. In what I have seen, it is more common to use \hat e_i for any orthonormal basis. See similar comment above about \vec b_i and \vec h_i in "Misuse of the Lame coefficients" Chris2crawford (talk) 12:51, 24 April 2023 (UTC)