Editing Mean value theorem (section)

== Mean value theorem in several variables ==
The mean value theorem generalizes to real functions of multiple variables. The trick is to use parametrization to create a real function of one variable, and then apply the one-variable theorem.

Let <math>G</math> be an open subset of <math>\R^n</math>, and let <math>f:G\to\R</math> be a differentiable function. Fix points <math>x,y\in G</math> such that the line segment between <math>x, y</math> lies in <math>G</math>, and define <math>g(t)=f\big((1-t)x+ty\big)</math>. Since <math>g</math> is a differentiable function in one variable, the mean value theorem gives:

:<math>g(1)-g(0)=g'(c)</math>

for some <math>c</math> between 0 and 1. But since <math>g(1)=f(y)</math> and <math>g(0)=f(x)</math>, computing <math>g'(c)</math> explicitly we have:

:<math>f(y)-f(x)=\nabla f\big((1-c)x+cy\big)\cdot (y-x)</math>

where <math>\nabla</math> denotes a [[gradient]] and <math>\cdot</math> a [[dot product]]. This is an exact analog of the theorem in one variable (in the case <math>n=1</math> this ''is'' the theorem in one variable). By the [[Cauchy–Schwarz inequality]], the equation gives the estimate:

:<math>\Bigl|f(y)-f(x)\Bigr| \le \Bigl|\nabla f\big((1-c)x+cy\big)\Bigr|\ \Bigl|y - x\Bigr|.</math>

In particular, when <math>G</math> is convex and the partial derivatives of <math>f</math> are bounded, <math>f</math> is [[Lipschitz continuity|Lipschitz continuous]] (and therefore [[Uniform continuity|uniformly continuous]]).

As an application of the above, we prove that <math>f</math> is constant if the open subset <math>G</math> is connected and every partial derivative of <math>f</math> is 0. Pick some point <math>x_0\in G</math>, and let <math>g(x)=f(x)-f(x_0)</math>. We want to show <math>g(x)=0</math> for every <math>x\in G</math>. For that, let <math>E=\{x\in G:g(x)=0\}</math>. Then <math>E</math> is closed in <math>G</math> and nonempty. It is open too: for every <math>x\in E</math> ,

:<math>\Big|g(y)\Big|=\Big|g(y)-g(x)\Big|\le (0)\Big|y-x\Big|=0</math>

for every <math>y</math> in open ball centered at <math>x</math> and contained in <math>G</math>. Since <math>G</math> is connected, we conclude <math>E=G</math>.

The above arguments are made in a coordinate-free manner; hence, they generalize to the case when <math>G</math> is a subset of a Banach space.