# mean-value theorem for several variables

The mean-value theorem for a function of one real variable may be generalised for functions of arbitrarily many real variables; for the sake of concreteness, we here formulate it for the case of three variables:

Theorem.  If a function  $f(x,\,y,\,z)$  is continuously differentiable in an open set of $\mathbb{R}^{3}$ containing the points  $(x_{1},\,y_{1},\,z_{1})$  and  $(x_{2},\,y_{2},\,z_{2})$  and the line segment connecting them, then an equation

 $f(x_{2},\,y_{2},\,z_{2})-f(x_{1},\,y_{1},\,z_{1})\;=\;f^{\prime}_{x}(a,\,b,\,c% )(x_{2}\!-\!x_{1})+f^{\prime}_{y}(a,\,b,\,c)(y_{2}\!-\!y_{1})+f^{\prime}_{z}(a% ,\,b,\,c)(z_{2}\!-\!z_{1}),$

where $(a,\,b,\,c)$ an interior point of the line segment, is .

Title mean-value theorem for several variables MeanvalueTheoremForSeveralVariables 2013-03-22 19:11:36 2013-03-22 19:11:36 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 26A06 msc 26B05