lie derivative造句

• The Lie derivatives are represented by vector fields, as group of diffeomorphisms of " M " has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory.
• Where \ operatorname { Rc } is the Ricci curvature tensor, \ lambda \ in \ mathbb { R }, X is a vector field on M, and \ mathcal { L } represents the Lie derivative.
• The directional derivative of a function with respect to a vector field is one of the most fundamental concepts in differential geometry, and the Lie derivative of a function is simply defined to be the directional derivative of the function.
• The fact that these rods are physically " carried " along the trajectory means that they are " Lie-dragged ", or propagated so that the Lie derivative of each rod along the tangent vanishes.
• In terms of Lie derivatives, \ langle \ partial _ \ mu n | n \ rangle = \ langle n | \ partial _ \ mu n \ rangle = 0 according to the definition of the connection for the vector bundle.
• The Lie derivative with respect to a vector field is an "'R "'- derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold.
• By applying the preceding ideas to the local 1-parameter group of diffeomorphisms defined by a vector field on " M ", and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained.
• That is to say, we may generate the " r "-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, " r " times.
• :: * 2 . " The more modern approach uses all the tools and concepts of modern differential geometry-tangent spaces, the calculus of forms, Lie derivatives, and so on . " talk ) 22 : 32, 14 August 2016 ( UTC)
• Note that the antisymmetrized covariant derivative, and the Lie derivative " L " " u " " v " differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization " is " the Lie derivative.