What are curvilinear coordinates?

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.

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Some articles on 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 ...
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 described ... accurately as a differentiable manifold." Instead of Cartesian coordinates, when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols appear in the ... viewpoint of an inertial frame of reference in curvilinear coordinates ...
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 ... generalization 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 ... distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics ...
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 ... 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
... bound by a potential energy surface (PES) (or a force field) which is a function of 3N-6 coordinates ... describing 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 ... It is possible to write the quantum mechanical kinetic energy operator for such curvilinear coordinates, but it is hard to formulate a general theory applicable to any molecule ...