The multi-dimensional Faa di Bruno formula

Abstract

The formula for higher order derivatives of a composite function of one variable was given by Faa di Bruno in 1855. Yamanaka gave a chain rule in the sense of Frechet derivative with respect to infinite dimensional variable. Fujiwara gave a chain rule for the case of two real variables. In this paper, a direct generalization of the Faa di Bruno formula to the several variable case was given. The multi-dimensional Faa di Bruno formula, that is the main result of this paper, was proved. In various problems on Partial Differential Equations (PDE), the higher order derivatives of composite functions of the form f(g(sub l)(x,t), ..., gm(sub m)(x,t),x,t) rather than the form f(g(x)) have to be calculated. Therefore a chain rule for such composite functions was proved. The modified formula is expected to be useful when one solves the Cauchy problem for a non linear PDE.

一変数合成関数の高階微分公式は、1855年にFaa di Brunoにより与えられた。Yamanakaはフレシエ微分の意味での無限次元チェインルールを、Fujiwaraは実二変数のチェインルールを与えた。本稿ではFaa di Bruno公式の直接的な実多変数公式を与えた。本稿の主要結果である、多次元Faa di Bruno公式を証明した。偏微分方程式の問題では、f(g(x))型ではなく、f(g(sub l)(x,t), ..., g(sub m)(x,t),x,t)型の合成関数の高次微分を計算する必要がある。そこでこの型の合成関数に対するチェインルールを求めた。これは非線形偏微分方程式のコーシー問題の解決や関連する解析学において有益である。