Coordinate surface
The coordinate surfaces of a <math>D</math>-dimensional coordinate system are the <math>(D-1)</math>-dimensional surfaces in Cartesian coordinates on which a particular coordinate of the original system is held constant.
For example, the coordinate surfaces of the <math>r</math> coordinate of the spherical coordinate system are spheres
- <math>
x^{2} + y^{2} + z^{2} = r^{2}</math>
Similarly, the coordinate surfaces of the colatitude coordinate <math>\theta</math> are cones, and those of the azimuthal angle <math>\phi</math> are planes.
The coordinate unit vectors are the unit-vector normals perpendicular to the coordinate surfaces. They point in the direction (in Cartesian space) ofthe fastest increase of the corresponding coordinate. For example, the unit vector <math>\mathbf{e}_{r}</math> for the <math>r</math> coordinate of the spherical coordinate system points outwards radially, in the direction of increasing <math>r</math>.
Coordinate lines may be defined in two ways. Usually, they represent the one-dimensional curves whose tangent vectors are the coordinate unit vectors. However, they may also be defined as are the <math>(D-2)</math>-dimensional curves along which the coordinate surfaces intersect. For <math>D=3</math>,these two definitions agree. Thus, the coordinate line for the <math>r</math> coordinate of the spherical coordinate system is a ray that points outwards radially from the origin and that is the intersection of the coordinate surfaces for <math>\theta</math> (a cone) and <math>\phi</math> (a plane).
Categories
Coordinate systems