# 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 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.

### Other articles related to "curvilinear coordinates, coordinate, coordinates, curvilinear coordinate":

Fictitious Forces in General Curvilinear Coordinates
... An inertial coordinate system is defined as a system of space and time coordinates x1, x2, x3, t in terms of which the equations of motion of a particle free of external forces are simply d2xj/dt2 = 0 ... In this context, a coordinate system can fail to be “inertial” either due to non-straight time axis or non-straight space axes (or both) ... In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both ...
The GF Method
... are bound by a potential energy surface (PES) (or a force field) which is a function of 3N-6 coordinates ... the PES in an optimum way are often non-linear they are for instance valence coordinates, such as bending and torsion angles and bond stretches ... the quantum mechanical kinetic energy operator for such curvilinear coordinates, but it is hard to formulate a general theory applicable to any molecule ...
Mechanics Of Planar Particle Motion - Fictitious Forces in Curvilinear Coordinates
... See also Curvilinear coordinate system and Covariant derivative To quote Bullo and Lewis "Only in exceptional circumstances can the configuration of Lagrangian system be ... curved space, or more accurately as a differentiable manifold." Instead of Cartesian coordinates, when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols ... of a particle motion from the viewpoint of an inertial frame of reference in curvilinear coordinates ...
Unit Length - Curvilinear Coordinates
... In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors equal to the degrees of freedom of the space ...
Pythagorean Theorem - Consequences and Uses of The Theorem - Euclidean Distance in Various Coordinate Systems
... The distance formula in Cartesian coordinates is derived from the Pythagorean theorem ... of the Pythagorean theorem, as If Cartesian coordinates are not used, for example, if polar coordinates are used in two dimensions or, in more general terms, if curvilinear coordinates are used ... straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics ...