# Correct Notation of Mean Value Theorem for Vector-Valued Function

Let $${\bf f}({\bf x})$$ be a function $${\bf f}: \mathbb{R}^n \to \mathbb{R}^m$$ with continuous derivatives $${\bf H}({\bf x})$$. We wish to approximate $${\bf f}({\bf x}_0)$$ by $${\bf f}({\bf x})$$.

It is well known that for $$m > 1$$, we cannot guarantee the existence of a vector $$\bf \tilde{x}$$ between $${\bf x}_0$$ and $${\bf x}$$ such that

$${\bf f}({\bf x}) = {\bf f}({\bf x}_0) + {\bf H}({\bf \tilde{x}})({\bf x}-{\bf x}_0)$$

Hence, the MVT cannot be directly applied to vector-valued function. However, a straightforward modification to the MVT can yield fruitful results, but seems ignored by many sources (including this very site).

We can apply the MVT to each of the $$m$$ components of $${\bf f}$$ separately, and thus we can write

$${f_k}({\bf x}) = f_k({\bf x}_0) + {\bf h}_k({\bf \tilde{x}}_k)({\bf x}-{\bf x}_0) , k = 1,...,m$$

I have seen some authors combine the $$m$$ results and write

$${\bf f}({\bf x}) = {\bf f}({\bf x}_0) + {\bf H}({\bf {x}_*})({\bf x}-{\bf x}_0)$$

where $${\bf {x}_*} =[{\bf \tilde{x}}_1, ...,{\bf \tilde{x}}_m]^T$$ is now a $$m \times n$$ matrix, and $${\bf H}({\bf {x}_*}) = [{\bf h}_1({\bf \tilde{x}}_1),...,{\bf h}_m({\bf \tilde{x}}_m)]^T$$.

Although I remember seeing this notation multiple times, I can't remember how exactly how it was expressed, or in which article I viewed it.

Could someone kindly reminds me how to properly use this notation for vector-valued functions?

Given such an ${\bf f}$ with open domain $\Omega\subset{\mathbb R}^n$, let $\tilde\Omega:=\bigl\{({\bf x},{\bf y})\bigm| [{\bf x},{\bf y}]\subset\Omega\bigr\}$, whereby $[{\bf x},{\bf y}]$ denotes the segment connecting ${\bf x}$ with ${\bf y}$. Then there is a continuous function $L:\>\tilde\Omega\to{\cal L}({\mathbb R}^n,{\mathbb R}^m)$, the linear maps from ${\mathbb R}^n$ to ${\mathbb R}^m$, such that $${\bf f}({\bf y})-{\bf f}({\bf x})=L({\bf x},{\bf y}).({\bf y}-{\bf x})\qquad\forall\>({\bf x},{\bf y})\in\tilde\Omega\ .$$ Furthermore $$L({\bf x},{\bf x})=d{\bf f}({\bf x})\qquad\forall\,{\bf x}\in\Omega\ .$$ This $L({\bf x},{\bf y})$ is exactly what you denote by ${\bf H}({\bf x_*})$. But neither your ${\bf x}_*$ is a point, nor ${\bf H}$ is a function of one vector variable.
Proof. Fix $({\bf x},{\bf y})\in\tilde\Omega$, and consider the auxiliary function $$\phi(t):={\bf f}\bigl((1-t){\bf x}+t{\bf y}\bigr)\qquad(0\leq t\leq1)$$ with derivative $\phi'(t)=d{\bf f}\bigl((1-t){\bf x}+t{\bf y}\bigr).({\bf y}-{\bf x})$. One then has $${\bf f}({\bf y})-{\bf f}({\bf x})=\phi(1)-\phi(0)=\int_0^1\phi'(t)\>dt =\int_0^1 d{\bf f}\bigl((1-t){\bf x}+t{\bf y}\bigr).({\bf y}-{\bf x})\>dt=\left(\int_0^1 d{\bf f}\bigl((1-t){\bf x}+t{\bf y}\bigr)\>dt\right).({\bf y}-{\bf x})\ .$$ It follows that $$L({\bf x},{\bf y}):=\int_0^1 d{\bf f}\bigl((1-t){\bf x}+t{\bf y}\bigr)\>dt$$ has the stated properties.