User:Maschen/Curvilinear coordinates

In geometry, curvilinear coordinates are real numbers used to specify points in space, derived from lengths along straight lines, arc-lengths measured along curves, or angles. The simpler notions of a Cartesian coordinate system and orthogonal coordinates are included as special cases: Cartesian coordinates are all defined from straight lines all orthogonal to each other, with an orthonormal basis everywhere, while orthogonal coordinates are defined from curves which are always orthogonal at every point, with local orthonormal basis vectors. General curvilinear coordinates do not have to be orthogonal and the basis vectors do not have to be orthonormal.

Curvillinear coordinates are a very general formalism, applicable to pure space or spacetime, and flat or curved spaces (or spacetimes).

The formalism is important in continuum mechanics, when describing large nonlinear deformations in solids and fluids, and general relativity, for describing arbitary motion.

Definitions and geometric interpretations of covariant and contravariant bases and components
The formalism will be developed for flat 3d space first, and not 3 + 1 spacetime. Generalizations to higher dimensions and inclusion of time are given at the end.

Coordinate curves and surfaces
The coordinate curves are those curves which one coordinate varies and all the others are fixed constant. In 3d, these will be 1d space curves.

The coordinate surfaces are those curves which one coordinate is fixed constant and all the others can vary. In 3d, these will be 2d curved surfaces.

Tangent basis vectors can be found by taking the partial derivatives of the position vector along the coordinate curves:


 * $$\mathbf{e}_i = \frac{\partial\mathbf{r}}{\partial x^i}$$

and normal basis vectors can be found by taking the gradients of the coordinates:


 * $$\mathbf{e}^i = \nabla x^i $$

The tangent basis is referred to as a covariant basis, while the normal basis is a contravariant basis. These basis vectors are dual to one another, and are biorthogonal:


 * $$\mathbf{e}_i \cdot \mathbf{e}^j = \delta_i{}^j $$

Vector algebra
Addition of vectors remains streightforward to calculate, but can only be done at a point, since the basis vectors change. Vector multiplication like the dot, cross, and exterior products take more complicated forms.

Metric tensor
The metric tensor is required to provide the notion of length in an arbitrary coordinate system.

The fully covariant components are defined by:


 * $$\mathbf{e}^i \cdot \mathbf{e}^j = g^{ij} $$

the fully contravariant are defined by:


 * $$\mathbf{e}_i \cdot \mathbf{e}_j = g_{ij} $$

and the fully mixed components are defined by:


 * $$\mathbf{e}^i \cdot \mathbf{e}_j = g^i{}_j \,, \quad \mathbf{e}_i \cdot \mathbf{e}^j = g_i{}^j $$

Now we have the square of a magnitude (or norm or modulus) of a vector defined by:


 * $$|\mathbf{a}|^2 = \mathbf{a} \cdot \mathbf{a} = a_i\mathbf{e}^i \cdot a_j\mathbf{e}^j = a_i a_j \mathbf{e}^i \cdot \mathbf{e}^j = a_i a_j g^{ij} $$