Editing Curvilinear coordinates (section)

{{Short description|Coordinate system whose directions vary in space}}
{{redirect-distinguish|Lamé coefficients|Lamé parameters (solid mechanics)}}
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[[File:Curvilinear.svg|thumb|upright=1.25|<span style="color:blue">'''Curvilinear'''</span> (top), [[Affine coordinate system|<span style="color:red">'''affine'''</span>]] (right), and [[Cartesian coordinate system|<span style="color:black">'''Cartesian'''</span>]] (left) coordinates in two-dimensional space]]

In [[geometry]], '''curvilinear coordinates''' are a [[coordinate system]] for [[Euclidean space]] in which the [[coordinate line]]s may be curved. These coordinates may be derived from a set of [[Cartesian coordinate]]s by using a transformation that is [[invertible|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 [[Gabriel Lamé|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 ('''R'''<sup>3</sup>) are [[Cylindrical coordinate system|cylindrical]] and [[spherical coordinates|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, [[scalar (mathematics)|scalar]]s, [[vector (geometric)|vector]]s, or [[tensor]]s. Mathematical expressions involving these quantities in [[vector calculus]] and [[tensor analysis]] (such as the [[gradient]], [[divergence]], [[curl (mathematics)|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 force]]s is usually easier to solve in [[spherical coordinates]] than in Cartesian coordinates; this is true of many physical problems with [[Circular symmetry|spherical symmetry]] defined in '''R'''<sup>3</sup>.  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 is 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]], [[Theory of relativity|relativity]], and [[engineering]].