Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The namecurvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.
In two dimensional Cartesian coordinates, we can represent a point in space by the coordinates (x1,x2) and in vector form as where are basis vectors. We can describe the same point in curvilinear coordinates in a similar manner, except that the coordinates are now (ξ1,ξ2) and the position vector is . The quantities ξi and xi are related by the curvilinear transformation . The basis vectors and are related by
The coordinate lines in a curvilinear coordinate systems are level curves of ξ1 and ξ2 in the two-dimensional plane.
An example of a curvilinear coordinate system in two-dimensions is the polar coordinate system. In that case the transformation is
Other well-known examples of curvilinear systems are cylindrical and spherical polar coordinates for R3. While a Cartesian coordinate surface is a plane, e.g., z = 0 defines the x–y plane, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere in R3—which obviously is curved.
Coordinates are often used to define the location or distribution of physical quantities which may be scalars, vectors, or tensors. Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R3 (e.g., motion in the field of a point mass/charge), is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Also boundary conditions may enforce symmetry. One would describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere.
Many of the concepts in vector calculus, which are given in Cartesian or spherical polar coordinates, can be formulated in arbitrary curvilinear coordinates. This gives a certain economy of thought, as it is possible to derive general expressions, valid for any curvilinear coordinate system, for concepts such as the gradient, divergence, curl, and the Laplacian.