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 3 containing the points  (x1,y1,z1)  and  (x2,y2,z2)  and the line segmentMathworldPlanetmath connecting them, then an equation


where (a,b,c) an interior pointPlanetmathPlanetmath 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