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I'm having a lot of trouble deciphering the notation in this proof of the mean value theorem in several variables. For example, on page 2 of this link we see an example of why the multivariable mean value equality fails & a claim that the best we can do is to find an inequality, yet the pages I've posted provide an equality. I can't understand the notation, & I'm afraid to learn something like this when a ton of sources clearly claim a mean value equality theorem doesn't hold, for example on stack, so I'd sincerely appreciate some help with this, like a comprehensive explanation by someone interested, because if this proof is valid then an extremely easy proof of the inverse function theorem analogous to the single-variable version follows automatically, many thanks!

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bolbteppa
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$\def \R {\mathbb{R}}$ It depends on what you mean by mean value theorem in several variables. What doesn't work is mean value theorem for $f: \R^n \to \R^m$ for $m > 1$ since each coordinate in codomain can dictate a different point in domain. But the case $m = 1$, $n > 1$ is ok.

The conterexample on page 2 of InvFT is a counterexample for following mean value theorem: For differentiable $f: \R^n \to \R^m$, there is a point $c$ on line $[a, b] ⊆ \R^n$ such that $f(b) - f(a) = (D_c f)(b - a)$, where $D_c f: \R^n \to \R^m$ is the derivative / differential of $f$ at point $c$. However this theorem holds if $m = 1$ as Theorem 9 from your scanned source shows.

But Theorem 11 on page 14 from your scanned source says something different. It says that there are points $c_i$ on line $[a, b]$, $i ∈ \{1, …, m\}$ such that $f(b) - f(a) = L(b - a)$ where $L = [D_{c_i} f_i: i ∈ \{1, …, m\}]$ where $f_i$ is $i$-th component of $f$.

user87690
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  • Page 14 that I posted claims the opposite! – bolbteppa Nov 14 '13 at 14:45
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    @bolbteppa: Page 14 says the same what I said. Each coordinate in codomain can dictate different point in domain, so the mean value theorem in the sense “there exists one point in domain, derivative at which gives the difference” doesn't hold. Instead you have different point for each coordinate in the codomain and since each point itself has more coordinates, you can put all these coordinates into a matrix. – user87690 Nov 14 '13 at 15:01
  • Would you mind modifying your answer to explain how the theorem on page 14 applies to the counter-example to the multi-variable mean value theorem on page 2 of this: http://artsci.wustl.edu/~e511jn/InvFT.pdf ??? Just do not see the underlying thing going on here :( – bolbteppa Nov 14 '13 at 15:32
  • @bolbteppa: I've tried. – user87690 Nov 14 '13 at 16:12