Time dependent electric field: Mathematical expansion for local electric field In many articles and books I see that local electric field is expanded as 
$$\vec E_0(\vec r(t)) = \vec E_0(\vec R_0) − (\vec a(t) \cdot \nabla) \vec E_0(\vec R_0) \cos(\Omega t) + \ldots $$  
For example in the page 2, equation 3 on http://arxiv.org/pdf/0902.2746.pdf
I never understood this expansion. Can anyone please explain me what is this expansion and on what conditions we can express a local electric field in this form. What is the physical interpretation of this expansion?
 A: Note that $\mathbf r(t)$ is the trajectory (a priori unknown) of a charged particle in an external electric field. Now consider the ansatz $\mathbf r(t) = \mathbf r_0(t) - \mathbf a(t) \cos \Omega t$, which is motivated by the solution for a homogeneous electric field $\mathbf E(t) = \frac{m\Omega^2}{q} \mathbf a \cos \Omega t$. Here $\mathbf r_0(t)$ is a slow drift due to the spatial variation of the electric field and $\mathbf a(t) \cos \Omega t$ is a faster oscillatory motion due to the driving of the field (also present for a homogeneous field).
Every component of the electric field along this trajectory $E_i (\mathbf r(t))$ can be formally Taylor expanded at each instant $t$ near $\mathbf r_0(t)$ up to first order in $\mathbf a(t)$ as follows
\begin{align}
E_i ( \mathbf r(t) ) & \simeq E_i ( \mathbf r_0(t) ) + \left. \nabla E_i \right|_{\mathbf r(t) = \mathbf r_0(t)} \cdot \left( \mathbf r(t) - \mathbf r_0(t) \right) \\
& = E_i ( \mathbf r_0(t) ) - \mathbf a(t) \cdot \left. \nabla E_i \right|_{\mathbf r(t) = \mathbf r_0(t)} \cos \Omega t,
\end{align}
or in vector notation:
\begin{equation}
\mathbf E ( \mathbf r(t) ) \simeq \mathbf E ( \mathbf r_0(t) ) - \cos \Omega t\left[\left( \mathbf a(t) \cdot \nabla \right) \mathbf{E} \right](\mathbf r_0(t)).
\end{equation}
