second covariant derivative

Lorsqu'on fixe un système de coordonnées, cette dérivée se transforme lors d'un changement … The Covariant Derivative in Electromagnetism We’re talking blithely about derivatives, but it’s not obvious how to define a derivative in the context of general relativity in such a way that taking a derivative results in well-behaved tensor.

Cependant, une métrique définit de manière unique une dérivée covariante particulière appelée la,Les propriétés de la dérivée impliquent que,L'information sur le voisinage du point P peut être utilisée pour définir au moyen d'une dérivée covariante le,Une fois que la dérivée covariante est définie pour les champs de vecteurs, elle peut être étendue aux,La dérivée covariante d'un champ de tenseurs le long d'un vecteur,Le premier terme de cette formule décrit la « déformation » du système de coordonnées par rapport à la dérivée covariante, et le second les changements de coordonnées du vecteur.La dérivée covariante est la dérivée selon les coordonnées à laquelle on ajoute des termes correctifs décrivant l'évolution des coordonnées.La dérivée covariante d'un tenseur de type (r,s) par rapport à.Dans les ouvrages de physique, la dérivée covariante est parfois décrite à l'aide de ses composantes dans les équations.On note souvent la dérivée covariante à l'aide d'un,À nouveau, ceci montre que la dérivée covariante d'un champ vectoriel n'est pas simplement obtenue en dérivant les coordonnées,Dans d'autres textes plus anciens (incluant l'ouvrage,Parfois la dérivée covariante le long d'une courbe est qualifiée d'.Un article de Wikipédia, l'encyclopédie libre.https://fr.wikipedia.org/w/index.php?title=Dérivée_covariante&oldid=172899580,Article contenant un appel à traduction en anglais,licence Creative Commons attribution, partage dans les mêmes conditions,comment citer les auteurs et mentionner la licence.La dernière modification de cette page a été faite le 14 juillet 2020 à 13:18. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.The covariant derivative of a covector field along a vector field,Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary,The covariant derivative of a tensor field along a vector field,The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination,Then using the rules in the definition, we find that for general vector fields,The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field.In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.Or, in words: take the partial derivative of the tensor and add:If instead of a tensor, one is trying to differentiate a.where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. There are 14 second covariant derivative-related words in total (not very many, I know), with the top 5 most semantically related being covariant derivative, manifold, differential geometry, vector calculus and smoothness.You can get the definition(s) of a word … Similarly, one may also obtain the second covariant derivative of a function,Again, for the torsion-free Levi-Civita connection, and for any vector fields.That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.Covariant derivatives do not commute; i.e.The latter can be shown by taking (without loss of generality) that.In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.Often a notation is used in which the covariant derivative is given with a,Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates.In some older texts (notably Adler, Bazin & Schiffer,The derivative along a curve is also used to define the,Sometimes the covariant derivative along a curve is called.A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.There is however another generalization of directional derivatives which,Note that the antisymmetrized covariant derivative,Specification of derivatives along tangent vectors of a manifold,This article is about covariant derivatives.
For example, if the same covariant derivative is written in,Consider the example of moving along a curve,In a curved space, such as the surface of the Earth (regarded as a sphere), the,(Since the manifold metric is always assumed to be regular, the compatibility condition implies linear independence of the partial derivative tangent vectors. Un vecteur est un objet géométrique indépendant du choix de la base. Ask Question Asked 2 months ago. ),(using the symmetry of the scalar product and swapping the order of partial differentiations).and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Travel around the circle at a constant speed.

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