Lie Derivative

In mathematics, the Lie derivative ( /ˈliː/), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow of another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

Functions, tensor fields and forms can be differentiated with respect to a vector field. Since a vector field is a derivation of zero degree on the algebra of smooth functions, the Lie derivative of a function along a vector field is the evaluation, i.e., is simply the application of the vector field. The process of Lie differentiation extends to a derivation of zero degree on the algebra of tensor fields over a manifold M. It also commutes with contraction and the exterior derivative on differential forms. This uniquely determines the Lie derivative and it follows that for vector fields the Lie derivative is the commutator

It also shows that the Lie derivatives on M are an infinite-dimensional Lie algebra representation of the Lie algebra of vector fields with the Lie bracket defined by the commutator,

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Considering vector fields as infinitesimal generators of flows (active diffeomorphisms) on M, the Lie derivatives are the infinitesimal representation of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.

Generalisations exist for spinor fields, fibre bundles with connection and vector valued differential forms.