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
Charles E. M. Pearce
  School of Mathematical Sciences
  University of Adelaide
  Adelaide, SA 5005

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.
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