多元向量函数的中值定理及应用

Mean Value Theorem of Multivariate Vector Valued Functions and its Applications

中值定理是可微函数的重要性质,是证明某些等式和不等式的重要工具,而等式形式的向量函数的微分中值定理一般是不成立的,通常只能得到微分中值不等式.本文从一元函数的Newton-Leibniz公式出发,证明了一个多元向量函数等式形式的积分型中值定理.该定理揭示了多元向量函数等式形式的微分中值定理不成立的原因,也蕴含了微分中值不等式.

The mean value theorem is an important property of differentiable functions,which is an important tool to prove some equalities and inequalities.In this paper,we prove a mean value theorem of the multivariate vector valued function based on the Newton-Leibniz formula.At the same time,this theorem reveals the reason of the failure for the differential mean value theorem of multivariate vector valued functions,and also implies the differential mean value inequality.