Faa di Bruno's Formula for Generalizing the Chain Rule to Higher Derivatives An Analysis
This paper looks to present a proof of Faa di Bruno's formula as the method of proof (based on the multiplicative structure of higher order cotangent spaces) clarifies, in addition, the origin of the strange numerical coefficients and the orders of derivatives in the formula. Generalized Fa& di Bruno's formula an extension of the original Fait debunk formula to multivariate functionsis useful in different domains of analysis, statistics, and computational calculus; this for example, and for a discussion of the computational aspects, also see Since, several different proofs of this formula have been given, notably. Although higher order cotangent spaces are well known, they have not been used before to this end. Remark that the multiplicative structure of firstorder cotangent spaces vanish identically. Hence, this structure is only relevant to higher order. Divided differences are often considered as the coefficients of the Newton interpolating polynomial. Assume that x0,x1,,xn are distinct; then the divided differences of f are recursively. As an application of the generalized Fab diBruno's formula, we obtain a linear recurrence procedure to calculate higher order derivatives of the inverse mapping. First, of all, let us introduce some notations. Multi indices of length n are the elements a! (Qlr a, in Nn. The order of Q: is ICXJ cq (Ye. We set CY! crl! .a, and O! 1. Let us consider two open subsets U C_ Rn, V c Iw" and two differentiable maps of class. As in A homogeneous space is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, is isomorphic to the quotient space where is the isotropy group The choice of does not affect the isomorphism type of because all of the isotropy groups are conjugate. Many common spaces are homogeneous spaces, such as the hypersphere, A representation of a group is a group action of on a vector space by invertible linear maps. For example, the group of two elements has a representation by and A representation is a group homomorphism