Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Higher Chain Formula proved by Combinatorics. | Higher Chain Formula proved by Combinatorics Tsoy-Wo Ma School of Mathematics and Statistics University of Western Australia Nedlands . 6907 Australia. twma@ Submitted Dec 5 2008 Accepted Jun 13 2009 Published Jun 19 2009 Mathematics Subject Classifications 05A05 05A10 05A17 Abstract We present an elementary combinatorial proof of a formula to express the higher partial derivatives of composite functions in terms of those of factor functions. 1. Introduction. Consider h x G X c Rv -g G Y c R g G R where X Y are open subsets of Rv R respectively and f g are sufficiently smooth functions. Lots of work have been done to express the higher partial derivatives of the composite function h gf in terms of those of f g. Among many important contributions not included in our references see 1 2 3 and 4 for ụ V 1. See 5 for V 1 and any ụ. See 6 and 7 for ụ V 2. Finally see 8 9 10 11 and 12 for the general case. Here we propose a concise version in 6 with a combinatorial proof modified from 13 extending ụ from their one dimension to our many dimensions. This paper is self-contained and should be readable by broad audiences including students studying routine operations of partial differentiation. In addition to many other applications the corollary of this strategically important formula is required to investigate holomorphic functions on test spaces under coordinate transformations. 2. It is more intuitive to work with variables x x1 xv and y yi yM . For each k in the set Jn of integers 1 2 n let tk denote one of the independent variables x1 xv. A partition of Jn is a family of pairwise disjoint nonempty subsets of Jn whose union is Jn. Sets in a partition are called blocks. A block function is to assign a label to each block of a partition. The set of all functions from a partition P of Jn into J is denoted by P . The set of all partitions of Jn is denoted by Pn. 3. Lemma. dn z ifrT d Eì í TT tt d E. 1 ftir -n E Eln . z n y- B 1 n P C A P I R P A RJ I I RcP
