Special coordinate system in Differential Geometry
In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable (Busemann 1955).
and 25 Related for: Normal coordinates information
In differential geometry, normalcoordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate...
given time. Formally, normal modes are determined by solving a secular determinant, and then the normalcoordinates (over the normal modes) can be expressed...
obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normalcoordinates Qk. Normalcoordinates decouple the classical...
cryptography Normal bundle Normal cone, of a subscheme in algebraic geometry Normalcoordinates, in differential geometry, local coordinates obtained from...
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude...
Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating...
correspond to the X, Y, and Z coordinates, respectively, of the surface normal. In 1978 Jim Blinn described how the normals of a surface could be perturbed...
normalcoordinates (u, v) as ds2 = du2 + dv2 − K(u dv – v du)2/12 + …. Taking a coordinate change from normalcoordinates at p to normalcoordinates at...
standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): The polar angle...
the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. Nota bene:...
the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial...
geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from...
Fourier space which uses normal modes of the wavevector as variables instead coordinates of particles. The number of normal modes is same as the number...
geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively. At an arbitrary point P consider the line PN which is normal to...
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots...
normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution...
y)} in contrast to the normal ( y , x ) {\displaystyle (y,x)} atan2 input order. The opposite problem occurs when the coordinates (X1, Y1) of one point...
Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only. Other...
frame Frame bundle Inertial frame of reference Local coordinates Local spacetime structure Lorentz covariance Minkowski space Normalcoordinates v t e...
directional statistics, the projected normal distribution (also known as offset normal distribution or angular normal distribution) is a probability distribution...
called the nadir. The following are two independent horizontal angular coordinates: Altitude (alt.), sometimes referred to as elevation (el.) or apparent...
lines of geographical coordinates: East–west tangent to parallels, North–south tangent to meridians, and Up–down in the direction normal to the oblate spheroid...
of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates. Skew coordinates...