## 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:

Fictitious Forces in General

... An inertial

**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 ... 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 ...Mechanics Of Planar Particle Motion - Fictitious Forces in

... See also

**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 by a vector in a vector space ... 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 ... of a particle motion from 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

... The distance formula in Cartesian

**coordinates**is derived from the Pythagorean theorem ... 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 ... distance between two points is converted to**curvilinear coordinates**can be found in the applications of Legendre polynomials in physics ...The GF Method

... energy surface (PES) (or a force field) which is a function of 3N-6

... energy surface (PES) (or a force field) which is a function of 3N-6

**coordinates**... 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 ...Unit Length -

... In general, a

**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 ...