---
tag: QP8B74828C
kind: statement
role: proposition
name: Cartan's magic formula
msc: [58A10]
uses: [F45VQ47GPV, 0299VCW1TH, 0E3G0GRPZ6, 0ZHSTAG8ZY, PJWK0PKN6Q]
about: [PJWK0PKN6Q]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_derivative
---

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]], let $X$ be a smooth [[vector field|0299VCW1TH]] on $M$ and let $\omega$ be a smooth [[differential form|0E3G0GRPZ6]] on $M$.
Let $\mathcal{L}_X$ be the [[Lie derivative|PJWK0PKN6Q]] along $X$, and let $d$ be the [[exterior derivative|0ZHSTAG8ZY]].
Let $\iota_X$ be the interior product, given on a $k$-form by $(\iota_X \omega)(Y_1, \dots, Y_{k-1}) = \omega(X, Y_1, \dots, Y_{k-1})$.
Then
$$\mathcal{L}_X \omega = d(\iota_X \omega) + \iota_X(d\omega).$$

## 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.

## Explanations

### L1 human

The Lie derivative measures how a form changes when it is dragged along the flow of a vector field.
Cartan's formula computes this change from two simple operations: plugging the vector field into the form, and taking the exterior derivative.
It turns a question about flows into a short algebraic computation.

### L2 human

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.