Use this tag for questions about coordinate systems for Euclidean space for which coordinate lines may be curved.
In geometry, curvilinear coordinates are coordinate systems for Euclidean space for which coordinate lines may be curved. Commonly-used curvilinear coordinate systems include rectangular, polar, spherical, and cylindrical coordinate systems. Those coordinates may be derived from a set of Cartesian coordinates using a transformation that is locally invertible (a one-to-one map) at each point. That means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back.
In three-dimensional Euclidean space, familiar curvilinear coordinate systems are Cartesian, cylindrical, and spherical coordinates. A Cartesian coordinate surface is a coordinate plane; for example z = 0 defines the xy-plane. In the same space, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere.
Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be scalars, vectors, or tensors. Mathematical expressions—such as the gradient, divergence, curl, and Laplacian—involving those quantities in vector calculus and tensor analysis can be transformed from one coordinate system to another according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.
Curvilinear coordinates are typically used to simplify calculations or to aid the understanding of a problem. For example, motion of particles under the influence of central forces is usually easier to solve in spherical than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. One would for instance 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. Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as Earth sciences, cartography, quantum mechanics, relativity, and engineering.