How can I prove the n dim analogue of the m value theorem?
ps. sorry for the mistake
Let $\displaystyle D\subseteq \mathbb{R}^n$. Say that $\displaystyle \bold{a},\bold{b}\in D$ and the line segment between $\displaystyle \bold{a},\bold{b}$ is contained in $\displaystyle D$ i.e. $\displaystyle \bold{a}+t(\bold{b}-\bold{a}) \in D \mbox{ for }t\in [0,1]$. Now define $\displaystyle g(t) = f(\bold{a}+t(\bold{b}-\bold{a})$ and confirm that $\displaystyle g$ is a differentiable function on $\displaystyle (0,1)$ (by multi-variable chain rule) and continous on $\displaystyle [0,1]$ so by the basic mean value theorem we have $\displaystyle g'(c)= g(1) - g(0)$ but $\displaystyle g'(c) = \nabla f(\bold{c})\cdot (\bold{b}-\bold{a})$. That means, $\displaystyle \nabla f(\bold{c}) \cdot (\bold{b}-\bold{a}) = f(\bold{b})-f(\bold{a})$ for some $\displaystyle \bold{c}$ between the line segment. Q.E.D.