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
ℝ3 containing the points (x1,y1,z1) and (x2,y2,z2) and the line segment
connecting them, then an equation
| f(x2,y2,z2)-f(x1,y1,z1)=f′x(a,b,c)(x2-x1)+f′y(a,b,c)(y2-y1)+f′z(a,b,c)(z2-z1), |
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 |