Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Tangent vector
Résumé
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point is a linear derivation of the algebra defined by the set of germs at .
Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.
Let be a parametric smooth curve. The tangent vector is given by , where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter t. The unit tangent vector is given by
Given the curve
in , the unit tangent vector at is given by
If is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by or
then the tangent vector field is given by
Under a change of coordinates
the tangent vector in the ui-coordinate system is given by
where we have used the Einstein summation convention. Therefore, a tangent vector of a smooth curve will transform as a contravariant tensor of order one under a change of coordinates.
Let be a differentiable function and let be a vector in . We define the directional derivative in the direction at a point by
The tangent vector at the point may then be defined as
Let be differentiable functions, let be tangent vectors in at , and let . Then
Let be a differentiable manifold and let be the algebra of real-valued differentiable functions on . Then the tangent vector to at a point in the manifold is given by the derivation which shall be linear — i.e.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.