A mean value theorem in several variables

Abstract

We give a generalization of the mean value theorem to several variables that retains the structure of the original result. In one variable the mean value theorem says that if $g$ is continuous in a closed interval $[a,b]$ and the derivative $g^{\prime }(x)$ exists for all points $x\in (a,b)$ then there exists a point $\xi \in (a,b)$ such that $g^{\prime }(\xi )=\frac{g(b)-g(a)}{b-a}$. We first discuss a version of the idea of “being the derivative of a continuous function at a point” in several variables, the notion of proper values. Then we give a version of the mean value theorem in several variables.
We establish that if $g$ is continuous in a region of ${\mathbb{R}}^d$ that contains the $d$-rectangle $R$ given as $a_j\le x_j\le b_j$, $1\le j\le d$, and $f={\nabla }_1\cdots {\nabla }_{d}g$ is a locally integrable function with proper values at all interior points of $R$ then there exists a point $({\xi }_1,\dots ,{\xi }_d)\in \mathrm{Int}R$ such that

where $\mathfrak{V}$ is the set of vertices of the $d$-rectangle and $\sigma (\mathbf{v})$ is the number of indices $j$ for which $v_j=a_j$. In two variables it means that if the rectangle $R:a_1\le x_1\le b_1$, $a_2\le x_2\le b_2$, is contained in a region $U\subset {\mathbb{R}}^2$ and $f={\nabla }_1{\nabla }_{2}g$ is a locally integrable function with proper values at all points, then there exists a point $({\xi }_1,{\xi }_2)\in \mathrm{Int}R$ such that