Lie derivative of a differential form

Let $M$ be a smooth manifold, let $X$ be a smooth vector field with local flow $\varphi_t$ , meaning $\partial_t\varphi_t(p)=X_{\varphi_t(p)}$ and $\varphi_0(p)=p$ wherever defined near $t=0$ , and let $\omega\in\Omega^k(M)$ . The Lie derivative of $\omega$ along $X$ is

\mathcal{L}_X\omega \coloneqq \left.\frac{d}{dt}\right|_{t=0} \bigl(\varphi_t^*\omega\bigr),

where $\varphi_t^*$ denotes the pullback of differential forms.

Cartan’s formula

The Lie derivative can be computed without explicitly using the flow, via Cartan’s magic formula:

\mathcal{L}_X\omega = d(\iota_X\omega) + \iota_X(d\omega).

Here $\iota_X$ is the interior product (contraction) with $X$ , and $d$ is the exterior derivative.

Properties

For $\alpha\in\Omega^k(M)$ and $\beta\in\Omega^\ell(M)$ :

Degree 0 derivation with respect to the wedge product: $\mathcal{L}_X(\alpha\wedge\beta)=(\mathcal{L}_X\alpha)\wedge\beta+\alpha\wedge(\mathcal{L}_X\beta).$
Commutes with the exterior derivative: $\mathcal{L}_X(d\omega)=d(\mathcal{L}_X\omega).$
On functions: for $f\in C^\infty(M)$ , $\mathcal{L}_X f = X(f)$ .

Examples

Translation on $\mathbb{R}^n$ . On $\mathbb{R}^n$ , with $n\ge2$ , let $X=\partial/\partial x^1$ . For the 1-form $\omega=f(x)\,dx^2$ , $\mathcal{L}_X\omega = \frac{\partial f}{\partial x^1}\,dx^2,$ and in particular $\mathcal{L}_X(dx^2)=0$ .

Radial vector field scales the area form on $\mathbb{R}^2$ . On $\mathbb{R}^2$ with coordinates $(x,y)$ , let $X=x\,\partial_x+y\,\partial_y$ . For the standard 2-form $\mu=dx\wedge dy$ , one computes $\mathcal{L}_X\mu = 2\,\mu,$ reflecting that the flow of $X$ is dilation by $e^t$ , which scales area by $e^{2t}$ .

Rotations preserve the standard area form on $\mathbb{R}^2$ . Let $X=-y\,\partial_x + x\,\partial_y$ , whose flow is rotation. Then $\mathcal{L}_X(dx\wedge dy)=0,$ expressing invariance of the area form under rotations.