Research on the Generalization and Proof of Multivariable Differential Calculus in Multivariable Extreme Value Problems

Abstract

Multivariable differential calculus constitutes a cornerstone of mathematical analysis. Multivariable extremum problems, as critical applications of this discipline, permeate diverse fields including natural sciences, engineering technology, and economic management.
While single-variable function extremum determination methods are grounded in derivative theory, multivariable functionscharacterized by
increased independent variables and complex interdependenciesrequire extended applications of single-variable extremum theories. Building upon fundamental multivariable calculus principles, this study systematically generalizes extremum concepts and determination theorems
from single-variable frameworks. Through rigorous proofs of core theorems and validation of generalized results via practical examples, we
provide both theoretical foundations and practical references for solving multivariable extremum problems.