Coordinate systems

The mathematical description of a geometrical system (detector, magnetic field, etc.) can often be greatly simplified by expressing it in terms of an appropriate coordinate system.

Let be Cartesian (or Euclidean) coordinates. A point in space is represented by a vector

where is the origin and are Cartesian unit vectors. The line element is

Two different Euclidean coordinate systems are related by a translation and a rotation.

For more general coordinates uⁱ = uⁱ (x,y,z), the chain rule ( Jacobi Matrix)

gives the line element

where

Any vector defined at the point , e.g. an electric field , will be expressed in the old and the new system as

Thus,

The most important difference is that the new basis vectors vary with , while the Cartesian basis vectors are the same everywhere.

For orthogonal coordinate systems, which are the main systems in practice, one has g_ij = 0 for . It is then convenient to introduce orthonormal basis vectors at the point ,

An orthogonal matrix relates the two bases and ( Rotations),

The determinant |A| is everywhere either +1 (for a right-handed system) or -1 (left-handed), except possibly at singularities of the transformation. In two dimensions, the most common non-Cartesian system is that of polar coordinates. In three dimensions, the most commonly used, apart from Cartesian, cylindrical coordinates spherical coordinates.