Lie derivative

In differential geometry, the Lie derivative ( LEE), formulated by Władysław Ślebodziński and named after Sophus Lie, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by 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. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted L X T {\displaystyle {\mathcal {L}}_{X}T} . The differential operator T ↦ L X T {\displaystyle T\mapsto {\mathcal {L}}_{X}T} is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function. The Lie derivative of a vector field Y with respect to another vector field X is known as the "Lie bracket" of X and Y, and is often denoted [ X , Y ] {\displaystyle [X,Y]} instead of L X Y {\displaystyle {\mathcal {L}}_{X}Y} . The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity L [ X , Y ] T = L X L Y T − L Y L X T , {\displaystyle {\mathcal {L}}_{[X,Y]}T={\mathcal {L}}_{X}{\mathcal {L}}_{Y}T-{\mathcal {L}}_{Y}{\mathcal {L}}_{X}T,} valid for any vector fields X and Y and any tensor field T....