Generalization of the formula of Faa di Bruno for a composite function with a vector argument

Abstract

Abstract. The paper presents a new explicit formula for the nth total derivative of a composite function with a vector argument. The well-known formula of Faa di Bruno gives an expression for the nth 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 nth 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.

Chain rules for higher derivatives

“…It is surprising if anything new can be said about such a classical topic, but we have not found anything similar in [8] or other recent papers [2,9,11]. We needed the formula for the study of the spectral Carathéodory-Fejér problem: given k × k matrices V 0 , V 1 , .…”

On applications of Faà-di-Bruno formula

“…+ , (1.5) the scalar and vector forms of considered formula (1.1) coincides. By the linearity property this proves the general formula for a linear combination of exponential functions of form (1.5) and hence, for the general case; for the straightforward calculations see [3].…”

Computation of multiple Lie derivatives by algorithmic differentiation

“…Generalizations for vector-valued functions are given in [22,14]. Algorithmic differentiation allows an direct calculation of the Taylor coefficients z k for k = 0, .…”

“…Therefore, the coefficient of a second order univariate Taylor expansion cannot directly be used to compute the mixed derivative (23b). In particular, if we simply set s 1 = s 2 = t and x(t) = w(t, t), we obtain the Taylor coefficient (24) of (17) with x 1 = w 1 + w 2 and x 2 = w 11 + w 12 + w 22 , which is not the desired derivative (23b). In theory, derivative (23b) could be calculated from linear combinations of the vector y 2 for a set of appropriate Taylor coefficients of (22) as suggested in [10,12].…”