Curvilinear coordinates

From Glossary of Meteorology

curvilinear coordinates

Any linear coordinates that are not Cartesian coordinates.

If u, v, w are three functions of the Cartesian coordinates x, y, z and if at least one of these functions is not a linear combination of x, y, z, then u, v, w are curvilinear coordinates of the point the Cartesian coordinates of which are x, y, z provided that the Jacobian ∂(u,v,w)/∂(x,y,z) is not equal to zero. Any surface along which one of the three curvilinear coordinates is constant is called a coordinate surface; there are three families of such surfaces. Any line along which two of the three curvilinear coordinates are constant is called a coordinate line; there are three sets of such lines. Three distinct coordinate lines may be drawn through each point of space. The three straight lines each of which is tangent to one of the coordinate lines at a given point in space are called the local axes. If the local axes are everywhere mutually perpendicular, the curvilinear coordinates are said to be orthogonal or rectangular. Examples of frequently used curvilinear coordinates are polar coordinates and cylindrical coordinates.
See also natural coordinates, spherical coordinates.