Coordinate surface

The coordinate surfaces of a three dimensional coordinate system are the surfaces on which a particular coordinate of the system is constant, while the coordinate lines are the curves along which two of the coordinates of the system are constant. The coordinate lines are often associated with the coordinate which is not constant.

For example the coordinate surfaces of the "r" coordinate of the spherical coordinate system on Euclidean space are the spheres, given in Cartesian coordinates by:$x^\{2\}\; +\; y^\{2\}\; +\; z^\{2\}\; =\; R^\{2\}$for a fixed value "r"="R" of the "r" coordinate.

Similarly, the coordinate surfaces of the colatitude coordinate "θ" are cones, and those of the azimuthal angle "φ" are half-planes.

Thus, the coordinate "lines" for the "r" coordinate of the spherical coordinate system are the rays that point outwards radially from the origin: such a line is the intersection of coordinate surfaces for "θ" (a cone) and "φ" (a half-plane).

The coordinate unit vectors are the unit-vector normals perpendicular to the coordinate surfaces. They point in the direction of the fastest increase of the corresponding coordinate. For example, the unit vector $mathbf\{e\}\_\{r\}$ for the "r" coordinate of the spherical coordinate system points outwards radially, in the direction of increasing "r".

Coordinate lines are also defined for two dimensional coordinate systems: the coordinate lines "for" one of the two coordinates are the coordinate lines "of" the other coordinate, i.e., the curves along which the latter coordinate is constant.

These ideas generalize easily to coordinate systems in any dimension "D": the coordinate hypersurfaces are the hypersurfaces (typically submanifolds of dimension "D"−1) on which one of the coordinates is constant.

More generally, for any integer "k" between "1" and "D", the coordinate "leaves" of codimension "k" (typically submanifolds of dimension "D"−"k") are obtained by holding "k" of the coordinates in the coordinate system constant. When "D"="k"+1 or "D"="k"+2, these may again be called "coordinate lines" or "coordinate surfaces" (respectively).