lie derivative造句

• The Lie derivative is another derivative that is covariant, but which should not be confused with the " covariant derivative ".
• The Lie derivatives of general geometric objects ( i . e ., sections of natural fiber bundles ) were studied by K . Yano.
• The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between differential forms and Lie derivatives.
• There are many different forms of "'Lie bracket "', in particular the Lie derivative and the Jacobi Lie bracket.
• The Hamiltonian density is related to the Lie derivative of the Lagrangian density with respect to a unit timelike horizontal vector field on the gauge bundle.
• The Lie derivative of a type tensor field T along ( the flow of ) a contravariant vector field X ^ \ rho may be expressed as
• One of the main uses of the Lie derivative in general relativity is in the study of spacetime symmetries where tensors or other geometrical objects are preserved.
• These arise in the study of M . Gerstenhaber, Kunihiko Kodaira, and D . C . Spencer, and in the theory of Lie derivatives.
• A generalization of the Lie bracket is the Lie derivative, which allows differentiation of any tensor field along the flow generated by " X ".
• Here is the-valued function obtained by duality from pairing the one-form with the vector field, and is the Lie derivative of this function along.
• This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the curvature tensors built from them are.
• The Lie derivative is usually denoted by \ scriptstyle \ mathcal L _ X, where \ scriptstyle X is the vector field along whose congruence the Lie derivative is taken.
• The Lie derivative is usually denoted by \ scriptstyle \ mathcal L _ X, where \ scriptstyle X is the vector field along whose congruence the Lie derivative is taken.
• He studied applications of the Lie derivative as it relates to Riemannian geometry as well as absolute differential calculus, and published a large number of papers relating to the subjects.
• For some smooth real-valued function \ varphi on M, where \ mathcal { L } _ X g denotes the Lie derivative of the metric g with respect to X.
• The Lie derivative of a type relative tensor field \ Lambda of weight w \, along ( the flow of ) a contravariant vector field X ^ \ rho may be expressed as
• Together with the interior product ( a degree-1 derivation on the exterior algebra defined by contraction with a vector field ), the exterior derivative and the Lie derivative form a Lie superalgebra.
• An important notion here is the Lie derivative, \ mathcal { L } _ V, which is basically a derivative operation that infinitesimally " shifts " functions along some orbit with tangent vector V.
• This identity is known variously as " Cartan's formula " or " Cartan's magic formula ", and can be used as a definition of the Lie derivative of a differential form.
• 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.