The multivariate Faa di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

ANZIAM J. 48 (2007), 327-341

The multivariate Faà di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

Roy B. Leipnik
(Deceased October 10, 2006)
Department of Mathematics
UCSB, CA 93106–3080
USA

and

Charles E. M. Pearce
School of Mathematical Sciences
University of Adelaide
Adelaide, SA 5005
Australia
charles.pearce@adelaide.edu.au

Abstract

The Faà di Bruno formulae for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial literature on the univariate case, but despite significant applications the multivariate case has until recently received limited study. We present a succinct result which is a natural generalization of the univariate version. The derivation makes use of an explicit integral form of the remainder term for multivariate Taylor expansions.

Download the article in PDF format (size 151 Kb)

Australian Mathematical Publishing Association Inc.