# The Lie derivative of a vector field

Without some kind of additional structure, there is no way to “transport” vectors, or compare them at different points on a manifold, and therefore no way to construct a vector derivative. The simplest way to introduce this structure is via another vector field, which leads us to the Lie derivative $${L_{v}w\equiv\left[v,w\right]}$$; as noted above, $${L_{v}}$$ is a derivation on $${\mathrm{vect}(M)}$$ due to the Jacobi identity. In this section we define the Lie derivative in terms of infinitesimal vector transport, and explore its geometrical meaning.

Given any vector field $${v}$$ on $${M^{n}}$$, it can be shown ([18] p. 37) that there exists a locally unique parameterized curve $${v_{p}(t)}$$ at every point $${p\in M}$$ such that $${v_{p}(0)=p}$$ and $${\dot{v}_{p}(t)}$$ is the value of the vector field $${v}$$ at the point $${v_{p}(t)}$$ (the dot indicates the derivative with respect to $${t}$$, which as usual is calculated on the curve mapped to $${\mathbb{R}^{n}}$$ by the coordinate chart). Each curve in this family is in general only well-defined locally, i.e. for $${-\varepsilon<t<\varepsilon}$$, and is thus called the local flow of $${v}$$.

For a fixed value of $${t}$$, there is some region $${U\subset M}$$ where the map $${\Phi_{t}\colon U\to U}$$ defined by $${p\mapsto v_{p}\left(t\right)}$$ on all of $${M}$$ is a diffeomorphism, and within the valid domain of $${t}$$ the maps $${\Phi_{t}}$$ satisfy the abelian group law $${\Phi_{t}\circ\Phi_{s}=\Phi_{t+s}}$$; thus the $${\Phi_{t}}$$ are called a local one-parameter group of diffeomorphisms. This name is somewhat misleading, since due to the limited valid domain of $${t}$$ the maps $${\Phi_{t}}$$ do not actually form a group; the “local” reflects the fact that the diffeomorphisms are not on all of $${M}$$. In the case that these maps are in fact valid for all of $${t}$$ and $${M}$$, $${v}$$ is called a complete vector field, and the $${\Phi_{t}}$$ are called a one-parameter group of diffeomorphisms. If $${M}$$ is compact, then every vector field is complete; if not, then a vector field is complete if it has compact support (is zero except on a compact subset of $${M}$$).

The tangent map $${\mathrm{d}\Phi}$$ defined by the vector field $${v}$$ is then the extra structure we need to “transport” vectors. $${\mathrm{d}\Phi}$$ maps a vector tangent to the curve $${C}$$ to a vector tangent to the curve $${\Phi\left(C\right)}$$; it “pushes vectors along the flow of $${v}$$.” We can now define the Lie derivative as a limit

\begin{aligned}L_{v}w&\equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left[\mathrm{d}\Phi_{-\varepsilon}\left(w\left|_{v_{p}\left(\varepsilon\right)}\right.\right)-w\left|_{p}\right.\right]\\&=\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left[w\left|_{v_{p}\left(\varepsilon\right)}\right.-\mathrm{d}\Phi_{\varepsilon}\left(w\left|_{p}\right.\right)\right]\\&=\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left[w\left|_{p}\right.-\mathrm{d}\Phi_{\varepsilon}\left(w\left|_{v_{p}\left(-\varepsilon\right)}\right.\right)\right].
\end{aligned}

The Lie derivative $${L_{v}w}$$ is “the difference between $${w}$$ and its transport by the local flow of $${v}$$.”

 ◊ In this and future depictions of vector derivatives, the situation is simplified by focusing on the change in the vector field $${w}$$ while showing the “transport” of $${w}$$ as a parallel displacement. This has the advantage of highlighting the equivalency of defining the derivative at either 0 or $${\varepsilon}$$ in the limit $${\varepsilon\rightarrow0}$$. Depicting $${L_{v}w}$$ as a non-parallel vector at $${v_{p}\left(t\right)}$$ would be more accurate, but would obscure this fact. We also will follow the picture here in using words to characterize derivatives: namely, “the difference” is short for “the difference per unit $${\varepsilon}$$ to order $${\varepsilon}$$ in the limit $${\varepsilon\rightarrow0}$$.”

This definition can be shown to be equivalent to $${L_{v}w\equiv\left[v,w\right]}$$. Another way of depicting the Lie derivative that highlights the anti-commutativity of the Lie bracket is to consider $${L_{v}w}$$ in terms of a loop defined by the flows of $${v}$$ and $${w}$$, noting that $${v_{p}(\varepsilon)=p+\varepsilon v}$$.

The above depicts the Lie derivative $${L_{v}w}$$ as the vector field whose local flow is the “commutator of the flows of $${v}$$ and $${w}$$,” i.e. it is the difference between the local flow of $${v}$$ followed by $${w}$$ and that of $${w}$$ followed by $${v}$$. Thus $${L_{v}w}$$ “completes the parallelogram” formed by the flow lines.