I have convinced my self that in general Taylor's formula with mean value form of the remainder does not in extend to a multidimensional setting.
In a one-dimensional setting, we have that for a $f:\mathbb{R}^n \to \mathbb{R}$ with $f\in C^1$ that
$$
f(x)=f(z)+Df(\xi)(x-z),
$$
for any $x,z\in\mathbb{R}^n$ and some $\xi$ on the line segment between $x$ and $z$, i.e. $\xi= x+t(z-x)$, $t\in[0,1]$.
However, I have not been able to source a reference for a generalization of this theorem whenever $f:\mathbb{R}^n \to \mathbb{R}^m$ with $f\in C^1$. I know that I can use the one dimensional Taylor's formula for each of the coordinate functions, but as far as I can see, this will possibly result in different $\xi$'s for each coordinate function.
Is there a standard counterexample for the negation of the generalization? Have you encountered a reference investigating the generalization?
Best Answer
Note that the degree $1$ case of Taylor's theorem is the mean value theorem. In general, the mean-value theorem doesn't hold for vector-valued functions.