Editing Curvilinear coordinates (section)

=== Coordinates, basis, and vectors ===
[[File:General curvilinear coordinates 1.svg|thumb|upright=1.35|Fig. 1 - Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.]]
[[File:Spherical coordinate elements.svg|thumb|upright=1.35|Fig. 2 - Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. '''Surfaces:''' ''r'' - spheres, θ - cones, &Phi; - half-planes; '''Lines:''' ''r'' - straight beams, θ - vertical semicircles, &Phi; - horizontal circles;

'''Axes:''' ''r'' - straight beams, θ - tangents to vertical semicircles, &Phi; - tangents to horizontal circles]]

For now, consider [[three-dimensional space|3-D space]]. A point ''P'' in 3-D space (or its [[position vector]] '''r''') can be defined using Cartesian coordinates (''x'', ''y'', ''z'') [equivalently written (''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup>)], by <math>\mathbf{r} = x \mathbf{e}_x + y\mathbf{e}_y + z\mathbf{e}_z</math>, where '''e'''<sub>''x''</sub>, '''e'''<sub>''y''</sub>, '''e'''<sub>''z''</sub> are the ''[[standard basis]] vectors''.

It can also be defined by its '''curvilinear coordinates''' (''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>) if this triplet of numbers defines a single point in an unambiguous way. The relation between the coordinates is then given by the invertible transformation functions:

:<math> x = f^1(q^1, q^2, q^3),\, y = f^2(q^1, q^2, q^3),\, z = f^3(q^1, q^2, q^3)</math>
:<math> q^1 = g^1(x,y,z),\, q^2 = g^2(x,y,z),\, q^3 = g^3(x,y,z)</math>

The surfaces ''q''<sup>1</sup> = constant, ''q''<sup>2</sup> = constant, ''q''<sup>3</sup> = constant are called the '''coordinate surfaces'''; and the space curves formed by their intersection in pairs are called the '''[[coordinate curves]]'''. The '''coordinate axes''' are determined by the [[tangent]]s to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates.

In the Cartesian system, the standard basis vectors can be derived from the derivative of the location of point ''P'' with respect to the local coordinate

:<math>\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}.</math>

Applying the same derivatives to the curvilinear system locally at point ''P'' defines the natural basis vectors:

:<math>\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}.</math>

Such a basis, whose vectors change their direction and/or magnitude from point to point is called a '''local basis'''. All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are '''global bases''', and can be associated only with linear or [[affine coordinate system]]s.

For this article '''e''' is reserved for the [[standard basis]] (Cartesian) and '''h''' or '''b''' is for the curvilinear basis.

These may not have unit length, and may also not be orthogonal.  In the case that they ''are'' orthogonal at all points where the derivatives are well-defined, we define the '''[[#Relation to Lamé coefficients|Lamé coefficients]]'''{{anchor|Lamé coefficients}} (after [[Gabriel Lamé]]) by

:<math>h_1 = |\mathbf{h}_1|; \; h_2 = |\mathbf{h}_2|; \; h_3 = |\mathbf{h}_3|</math>

and the curvilinear [[orthonormal basis]] vectors by

:<math>\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}.</math>

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 <math>\mathbb{R}^3</math> at ''P'', and so are local to ''P''.)

In general, curvilinear coordinates allow the natural basis vectors '''h'''<sub>i</sub> not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of [[physics]] and [[engineering]], particularly [[fluid mechanics]] and [[continuum mechanics]], require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities.  A discussion of the general case appears later on this page.