High-order total differential solution of multivariate function based on Yang Hui triangle

Abstract

Under the current study, there are fewer ways to solve the higher-order full differential of multivariate functions. This paper introduces Yanghui Triangle and the higher-dimensional form of Yanghui Triangle to study the field of higher-order full differential solution of multivariate functions, to summarize the general formulas and to improve the efficiency of operation. Based on the assumptions of infinite partial derivatives and continuity of partial derivatives, this paper first proves the correctness of the higher-order total differentiation formula for binary functions by mathematical induction and tests it with examples, and the result proves that the formula is valid and has good applicability. Then, the higher-order total differential formula for ternary functions and more multivariate functions are mathematically inducted and proved to obtain a generalized formula, which improves the computational efficiency. The generalized formulas studied in this paper assume that multivariate functions can have partial derivatives and the partial derivatives are continuous, which is consistent with the assumptions of full differential computation of multivariate functions in practical applications of science and engineering technology. However, for the case where the partial derivatives of multivariate functions are not continuous, it needs to be proved in future studies.