normal$53719$ - traduction vers Anglais
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normal$53719$ - traduction vers Anglais

SPECIAL COORDINATE SYSTEM IN DIFFERENTIAL GEOMETRY
Geodesic normal coordinates; Normal coordinate; Normal neighborhood

normal      
n. Senkrechte; Normal
normal force         
  • Figure 2: Weight (''W''), the frictional force (''F''<sub>''r''</sub>), and the normal force (''F''<sub>''n''</sub>) acting on a block. Weight is the product of mass (''m'') and the acceleration of gravity (''g'').
FORCE EXERTED ON AN OBJECT BY A BODY WITH WHICH IT IS IN CONTACT, AND VICE VERSA
Normal Force; Normal reaction
normale Kraft
normal component         
  • Illustration of tangential and normal components of a vector to a surface.
Tangential component; Normal component; Tangent component; Perpendicular component; Paralel component; Parallel component
normaler Komponent (üblicher Komponent oder Kraft)

Wikipédia

Normal coordinates

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).