Electronic Proceedings of the Ninth Annual International Conference on Technology in Collegiate MathematicsReno, Nevada, November 710, 1996Paper C023
Multivariable Faa di Bruno Formulas 
Roy B. Leipnik
Department of Mathematics
University of California
Santa Barbara, CA 93106
USA
Phone: (805) 8932738
Fax: (805) 8932385
leipnik@math.ucsb.edu
 Troy T. Reid
Department of Mathematics
University of California
Santa Barbara, CA 93106
USA
Phone: (805) 8932738
Fax: (805) 8932385
reid@math.ucsb.edu

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The Bruno product for
multiple integer sequences is key to the combinatorial Faa di Bruno
formula for the univariate
higherorder chain rule derived in 1850 (It can be found in some
European calculus text books).
This extremely useful formula (in statistics and physics) is normally
restricted to derivatives of order
four. For multivariate statistics and solid (or fluid) mechanics,
composite derivative problems in
two, three, or four variables are quite normal, and sometimes appear
in undergraduate contexts.
Master's degree students are often exposed to multivariable problems
involving second, third, or
even fourthorder partial derivatives of composite functions.
Mistakes and omissions are frequent
when the ordinary firstorder chain rule is iteratively applied.
Keyword(s): multivariable calculus