Lie derivative of a tensor field (${\mathcal{L}}_X$)

For a smooth vector field $X$, the Lie derivative is the tensor derivation determined by ${\mathcal{L}}_{X}f=X(f)$ and ${\mathcal{L}}_{X}Y=[X,Y]$. The identity $[X,fY]=f[X,Y]+X(f)Y$ permits its unique extension. On a differential one-form,On differential forms it agrees with the usual Lie derivative of a differential form and Cartan's magic formula.

For a smooth function $f$ and vector field $X$,on every tensor field, with $X\otimes df$ interpreted as the endomorphism $Y\mapsto Y(f)X$. Both sides vanish on functions; on vector fields this follows from $[fX,Y]=f[X,Y]-Y(f)X$. Both are contraction-compatible tensor derivations, so agreement on functions and vector fields proves equality on all tensor types. In particular ${\mathcal{L}}_{fX}\omega =f{\mathcal{L}}_X\omega +\omega (X)df$.