Lie differential

of a tensor field $Q$ 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.