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Homemean-value theorem for several variables

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# 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 valid.

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## Mathematics Subject Classification

26A06*no label found*26B05

*no label found*

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