In general, we define a derivation to be a linear map \({\mathcal{D}\colon \mathfrak{a}\to \mathfrak{a}}\) on an algebra \({\mathfrak{a}}\) that follows the Leibniz rule (AKA product rule)

\(\displaystyle \mathcal{D}(AB)=(\mathcal{D}A)B+A(\mathcal{D}B). \)

As noted previously, the set \({\mathrm{vect}(M)}\) of vector fields on a manifold form a Lie algebra; the Lie bracket operation with a fixed vector field \({\left[u,\;\right]}\) is then a derivation on this algebra, since the Leibniz rule

\(\displaystyle \left[u,\left[v,w\right]\right]=\left[\left[u,v\right],w\right]+\left[v,\left[u,w\right]\right] \)

is just the Jacobi identity.

For a graded algebra, e.g. the exterior algebra, the degree of a derivation is the integer \({c}\) where \({\mathcal{D}\colon\Lambda^{k}M\to\Lambda^{k+c}M}\). A graded derivation is defined to follow the graded Liebniz rule, e.g. for a \({k}\)-form \({\varphi}\),

\(\displaystyle \mathcal{D}\left(\varphi\wedge\psi\right)=\mathcal{D}\varphi\wedge\psi+\left(-1\right)^{kc}\varphi\wedge\mathcal{D}\psi. \)

If \({c}\) is odd, a graded derivation is sometimes called an anti-derivation (AKA skew-derivation).

An Illustrated Handbook