Lie differential
of a tensor field in the direction of a vector field X
The principal linear part of increment of Q under its transformation induced by the local one-parameter group \phi_t of transformations of the manifold generated by X. The Lie differential \delta_X Q of a tensor field Q in the direction of a vector field X is equal to (\mathcal{L}_X Q)dt, where \mathcal{L}_X Q is the Lie derivative of Q in the direction of X.
The concept of a Lie differential admits the following physical interpretation. If a one-parameter transformation group \phi_t of a domain of the Euclidean space describes the stationary flow of a liquid with velocity field X, t is time and Q is a tensor field that describes some characteristic of the liquid (the deformation velocity tensor, stress tensor, density, etc.), then the Lie differential \delta_X Q describes the principal linear part of variation with time of Q from the point of view of an observer moving with the liquid, that is, in Lagrange variables.
Lie differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_differential&oldid=35921