In geometry, 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 name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.
Well-known examples of curvilinear systems are Cartesian, cylindrical and spherical polar coordinates, for R3, where R is the 3d space of real numbers.
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 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 of particles in a field), 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. Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as Earth sciences, cartography, and physics (in particular quantum mechanics, relativity), engineering, etc.
The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. General expressions in vector calculus and tensor analysis (such as the gradient, divergence, curl, Laplacian, and generalizations therefrom), valid for any curvilinear coordinate system, can be transformed to any coordinate system according to transformation rules and tensors.
Read more about Curvilinear Coordinates: Generalization To n Dimensions, Vector and Tensor Algebra in Three-dimensional Curvilinear Coordinates, Vector and Tensor Calculus in Three-dimensional Curvilinear Coordinates, Fictitious Forces in General Curvilinear Coordinates