Cartan's magic formula

Remarks

The idea of the proof: both sides are derivations of degree $0$ of the algebra of forms, and both commute with $d$. On a function $f$ both sides give $X f$, since $\iota_X f = 0$ and $\iota_X df = Xf$. On an exact $1$-form $df$ both sides give $d(Xf)$. Locally every form is a sum of products $f\, dg_1 \wedge \dots \wedge dg_k$, so the two sides agree everywhere.

Example: on $\R^2$, let $X = -y\, \partial_x + x\, \partial_y$, the infinitesimal rotation, and $\omega = dx \wedge dy$. Then $d\omega = 0$ and $\iota_X \omega = -y\, dy - x\, dx$, so $d(\iota_X \omega) = 0$. Hence $\mathcal{L}_X \omega = 0$: rotations preserve area.

The formula has two quick consequences. First, $\mathcal{L}_X$ commutes with $d$, since $d \mathcal{L}_X = d \iota_X d = \mathcal{L}_X d$ by $d^2 = 0$. Second, if $\omega$ is closed, then $\mathcal{L}_X \omega = d(\iota_X \omega)$ is exact.

Going deeper

For a student

The second consequence above is the heart of the homotopy invariance of de Rham cohomology. Integrating $\mathcal{L}_X$ along a flow shows that the pullbacks of a closed form by homotopic maps differ by an exact form. This gives the Poincaré lemma for star-shaped domains. In symplectic geometry the formula shows that a vector field $X$ preserves a symplectic form $\omega$ exactly when $\iota_X \omega$ is closed, which leads to Hamiltonian vector fields.