Curvilinear coordinates

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 coordinate systems in three-dimensional Euclidean space (R3) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.

Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) 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.

A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R3. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, it’s easier to describe the motion in a sphere with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.

Curvilinear (top), affine (right), and Cartesian (left) coordinates in two-dimensional space

1 Orthogonal curvilinear coordinates in 3 dimensions

1.1 Coordinates, basis, and vectors

2 Vector calculus

2.1 Differential elements

3 Covariant and contravariant bases

4 Integration

4.1 Constructing a covariant basis in one dimension

4.2 Constructing a covariant basis in three dimensions

4.3 Jacobian of the transformation

5 Generalization to n dimensions

6 Transformation of coordinates

7 Vector and tensor algebra in three-dimensional curvilinear coordinates

8 Tensors in curvilinear coordinates

8.1 The metric tensor in orthogonal curvilinear coordinates

8.1.1 Relation to Lamé coefficients

8.1.2 Example: Polar coordinates

8.2 The alternating tensor

8.3 Christoffel symbols

8.4 Vector operations

9 Vector and tensor calculus in three-dimensional curvilinear coordinates

9.1 Geometric elements

9.2 Integration

9.3 Differentiation

10 Fictitious forces in general curvilinear coordinates

11 See also