International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 7, Pages 481-491
Generalization of the formula of Faa di Bruno for a composite function
with a vector argument
Str. Asen Zlatarov 32A, Plovdiv 4000, Bulgaria
Received 12 January 1999
Copyright © 2000 Rumen L. Mishkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper presents a new explicit formula for the th total
derivative of a composite function with a vector argument. The
well-known formula of Faa di Bruno gives an expression for the
th derivative of a composite function with a scalar argument.
The formula proposed represents a straightforward generalization of
Faa di Bruno's formula and gives an explicit expression for the
th total derivative of a composite function when the argument is
a vector with an arbitrary number of components. In this sense, the
formula of Faa di Bruno is its special case. The mathematical tools
used include differential operators, polynomials, and Diophantine
equations. An example is shown for illustration.