8
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ See mathoverflow.net/questions/362718/… which is much simpler than the TAMS reference in Joe's answer. If one treats inner (for $G$) and outer (for $f$) derivatives using multiindex notation which is a form of decategorification one gets rather unreadable formulas. The simplest formulas follow from using multiindices for inner derivatives, but not for the outer ones which should instead be handled with tensor indices. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Dec 16, 2020 at 18:23

2 Answers 2

Reset to default
9
$\begingroup$

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:

Improve this answer
$\endgroup$
3
  • $\begingroup$ No I am not. The version that you are linking has $f : \mathbb R \rightarrow \mathbb R$, a function in a single variable. $\endgroup$
    – shuhalo
    Commented Dec 16, 2020 at 17:22
  • 3
    $\begingroup$ You're right, sorry about that. I would still suggest that what you want is a multivariate version of the formula of Faa di Bruno, but a version that's even more multi-variate than the one that happens to be in Wikipedia. My primary reason for posting an answer was to flag the phrase "Formula of Faa di Bruno" as a good starting search phrase to find what you want. (I speak as someone who wanted a reference for the 1-variable version a few years ago and was given this advice.) $\endgroup$
    – Joe Silverman
    Commented Dec 16, 2020 at 17:38
  • 3
    $\begingroup$ Not really relevant but I can't resist mentioning that Faa di Bruno was beatified in 1998. $\endgroup$
    – terceira
    Commented Oct 5, 2022 at 19:50
2
$\begingroup$

See page 8 of this paper

(You can replace the $i^{\text{th}}$ component of the function $F:\mathbb{R}^n \to \mathbb{R}^n$ with a function $f:\mathbb{R}^n \to \mathbb{R}$). It states that the derivative you are after may be described in terms of a `multicolour partition'.

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ Disclaimer: you are an author of that article. $\endgroup$
    – Alex M.
    Commented Oct 5, 2022 at 17:12
  • 3
    $\begingroup$ @Alex does it help now the full reference is there? $\endgroup$
    – David Roberts ♦
    Commented Oct 5, 2022 at 23:33

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .