# 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

$$f({x}_{2},{y}_{2},{z}_{2})-f({x}_{1},{y}_{1},{z}_{1})={f}_{x}^{\prime}(a,b,c)({x}_{2}-{x}_{1})+{f}_{y}^{\prime}(a,b,c)({y}_{2}-{y}_{1})+{f}_{z}^{\prime}(a,b,c)({z}_{2}-{z}_{1}),$$ |

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

Title | mean-value theorem for several variables |
---|---|

Canonical name | MeanvalueTheoremForSeveralVariables |

Date of creation | 2013-03-22 19:11:36 |

Last modified on | 2013-03-22 19:11:36 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26A06 |

Classification | msc 26B05 |