|WikiProject Mathematics||(Rated C-class, Mid-importance)|
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.....
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 bi and bi basis
The basis vector and cannot be normalized if one wants to keep the very important dot product rule .
Indeed, two vectors of unit length and whose dot product is equal to one have necessarily the same direction (cos θ = 1), meaning that bi 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 , which is not correct.
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...
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.