Lie derivative - Definition and Overview

In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by

<math> [A,B] \equiv \mathcal{L}_A B = - \mathcal{L}_B A<math>

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M. Looking at it the other way round, the diffeomorphism group of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory.

Lie derivative of tensor fields

In differential geometry, if we have a differentiable tensor T of rank (p q) (i.e. a differentiable linear map of smooth sections,

α, β, ...

of the cotangent bundle T*M and

X, Y, ...

of the tangent bundle TM,

T(α,β,...,X,Y,...)

such that for any smooth functions

f₁,...,f_p,...,f_p+q, T(f₁α,f₂β,...,f_p+1X,f_p+2Y,...)=f₁f₂...f_p+1f_p+2...f_p+qT(α,β,...,X,Y,...))

and a differentiable vector field (section of the tangent bundle) A , then the linear map

(£_AT)(α,β,...,X,Y,...)≡∇_A T(α,β,...,X,Y,...)-∇_{T(-,β,...,X,Y,...)}A(α)-...+ T(α,β,...,∇_XA,Y,...)+...

is independent of the connection ∇ used; as long as it is torsion-free, and in fact, is a tensor. This tensor is called the Lie derivative of T with respect to A.

In other words, if you have a tensor field T and an infinitesimal generator of a diffeomorphism given by a vector field U, then

£_UT

is nothing other than the infinitesimal change in T under the infinitesimal diffeomorphism.

Example Usage of derivative