Vector fields as differential operators

Vector fields $\vec{v}$ on a manifold (e.g., $\mathbb{R}^n$) can be identified with partial derivative operators

$$ \vec{v} \equiv \xi_1(x_1, \ldots, x_n) \frac{\partial}{\partial x_n} + \ldots + \xi_n(x_1, \ldots, x_n) \frac{\partial}{\partial x_n}, $$

or, more succinctly we may write,

$$ \vec{v} = \xi_1(\vec{x})\partial_{x_1} + \ldots + \xi_n(\vec{x})\partial_{x_n}; $$

this identification comes from the action of the vector field $\vec{v}$ on a function $f(\vec{x})$ (which can be real- or vector- or point-on-a-manifold-valued) by taking the directional derivative $D_{\vec{v}} f(\vec{x})$ along the vector field at each point $\vec{x}$ in its domain.

Lie series for values of functions along flows

One can consider how the function $f(\vec{x})$ behaves as $\vec{x}$ flows along the vector field $\vec{v}$; if flowing for a duration $t \in \mathbb{R}$, then we write the value of $f$ at the point to which $\vec{x}$ flows after that duration as $f(\exp(t\vec{v})\vec{x})$. We assume in this exposition that the coefficient functions $\xi_i, i = 1, \ldots, n$ and the function $f$ are entire (complex-analytic, in general) functions. Thus the flow commutes with $f$; we may therefore write $f(\exp(t\vec{v})\vec{x}0) = \left.\exp(t\vec{v})f(\vec{x})\right|{\vec{x}=\vec{x}_0}$.

It is possible to obtain an expansion of the value of $f(\vec{x})$ as $\vec{x}$ flows in the parameter $t$ along $\vec{v}$ as a power series in $t$:

$$ \left.\exp(t\vec{v})f(\vec{x})\right|{\vec{x}=\vec{x}0} = \left.\sum{n=0}^\infty \frac{t^n}{n!}\vec{v}^nf(\vec{x})\right|{\vec{x}=\vec{x}_0}, $$

where $\vec{v}^nf(\vec{x})$ is an $n$-fold (repeated) application of the directional derivative operator $\vec{v}$ to $f(\vec{x})$ and $\vec{v}^0f(\vec{x}) = f(\vec{x})$. This series expansion is called the Lie series [2] for $\exp(t\vec{v})f(\vec{x}_0)$.

Solving differential equations

With the coordinate vector function $f(\vec{x}) = \vec{x}$, this series is

$$ \exp(t\vec{v})\vec{x}0 = \left.\sum{n=0}^\infty \frac{t^n}{n!}\vec{v}^n\vec{x}\right|_{\vec{x}=\vec{x}_0}, $$

yielding a power series for the flow itself, along $\vec{v}$. This can be used to obtain an expansion of the solutions to any autonomous (or forced, but we do not consider this here) system of ordinary differential equations

$$ \dot{\vec{x}} = g(\vec{x}) $$

Since $g$ is $\mathbb{R}^n$-valued, we can write $g(\vec{x}) = g_1(\vec{x}) + \ldots + g_n(\vec{x})$, where $g_i$ is real-valued for all $i$. We may therefore view $\vec{x}(t)$ as the flow $\exp(t\vec{v})\vec{x}_0$ of $\vec{x}_0 = \vec{x}(0)$ in the parameter $t$ along the vector field