My question regards Muphrid's answer in this post.
He explains how
$$\Phi'(0)=\textbf{a}\nabla\phi(r_0),$$
but I get lost when he claims that the result holds "similarly for higher derivatives". I do not know how the result extends to higher derivatives of $\Phi$ nor why it extends in such a way.
Lastly, what is the meaning of
$$[\textbf{a}\nabla][\textbf{a}\nabla]\phi(r_0)\text{?}$$
