# Relationships between derivations

We can define one other derivation on $${k}$$-forms, the interior derivative (AKA inner derivative, inner multiplication), which is the generalization of the interior product to forms on manifolds; i.e. for a given vector $${v}$$ it is the graded degree $${-1}$$ derivation $${\left(i_{v}\varphi\right)\left(w_{2},\dotsc,w_{k}\right)\equiv\varphi\left(v,w_{2},\dotsc,w_{k}\right)}$$ on $${k}$$-forms $${\varphi}$$, which follows the graded Leibniz rule $${i_{v}\left(\varphi\wedge\psi\right)=(i_{v}\varphi)\wedge\psi+\left(-1\right)^{k}\varphi\wedge(i_{v}\psi)}$$. The graded commutativity of forms immediately gives the property $${i_{v}i_{w}+i_{w}i_{v}=i_{v}^{2}=0}$$.

The interior, exterior, and Lie derivatives then form an infinite-dimensional graded Lie algebra with the following relations:

• $${\left[L_{v},L_{w}\right]\equiv L_{v}L_{w}-L_{w}L_{v}=L_{\left[v,w\right]}}$$
• $${\left[i_{v},i_{w}\right]\equiv i_{v}i_{w}+i_{w}i_{v}=0}$$
• $${\left[\mathrm{d},\mathrm{d}\right]\equiv \mathrm{d}^{2}+\mathrm{d}^{2}=0}$$
• $${\left[L_{v},i_{w}\right]\equiv L_{v}i_{w}-i_{w}L_{v}=i_{\left[v,w\right]}}$$
• $${\left[L_{v},\mathrm{d}\right]\equiv L_{v}\mathrm{d}-\mathrm{d}L_{v}=0}$$
• $${\left[i_{v},\mathrm{d}\right]\equiv i_{v}\mathrm{d}+\mathrm{d}i_{v}=L_{v}}$$

This last relation is sometimes called Cartan’s formula (AKA Cartan’s magic formula).