TITLE: Multivariable higher-order chain rule
QUESTION [5 upvotes]: I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what is the derivative
$\partial^\alpha ( f \circ G )$
where $\alpha$ is a multiindex in the variables $x_1,\dots,x_m$ of degree $k$? We assume all necessary derivatives exist. References to the literature would be helpful too. I haven't found this general case treated in my sources.

REPLY [8 votes]: You’re looking for the multivariate version of the formula of Faa di Bruno.
Addendum: As the OP notes, the version in Wikipedia is not in sufficient generality, since it takes $f:\mathbb R\to\mathbb R$. For a version that allows $f$ to depend on more than one variable, see for example the article:

A  Multivariate  Faa  Di  Bruno  Formula with  Applications, M. Constantine  and  T.  H.  Savits, Trans. AMS, 348, Number 2, February 1996.