Let $K\in\mathbb{R}^*$ and $F:\mathbb{R}^3\to\mathbb{R}$ be of class $C^{\infty}$ be such that $$a+u\mapsto F(a+u),$$
where $a\in\mathbb{R}^3$ is a fixed nonnull vector and $u\in\mathbb{R}^3$ is such that $|u|\le K$.
(In case my notation should be ambiguous, I am just saying that when you read $F(a+u)$ it means that $F$ is evaluated in $a+u$ where $a$ is a fixed vector in $\mathbb{R}^3$ and $u\in\mathbb{R}^3$ satisfies $|u|\le K$).
I would like to evaluate the quantity $$\sum_{i=1}^3 \left\vert\frac{\partial F}{\partial u_i}(a+u) - \frac{\partial F}{\partial u_i}(a)\right\vert.$$ The only thing I am trying so far is the following. By means of Mean Value Theorem, it should be $$\sum_{i=1}^3\left\vert\frac{\partial F}{\partial u_i}(a+u) - \frac{\partial F}{\partial u_i}(a)\right\vert\le \sum_{i, j=1}^3\left\vert \frac{\partial^2 F}{\partial u_i \partial u_j} (z) \cdot u \right\vert\le K \sum_{i, j=1}^3\left\vert \frac{\partial^2 F}{\partial u_i \partial u_j} (z) \right\vert,$$ for a vector $z\in [a, a+u]$. My first question is: does my argument hold true? I do not feel confident about how I applied the MVT.
Arrived at this point (if everything is correct), I would please ask you to suggest a way to estimate the quantity $$\sum_{i, j=1}^3\left\vert \frac{\partial^2 F}{\partial u_i \partial u_j} (z) \right\vert.$$
My idea was to estimate with $$\max_{-K\le v\le K}\left\vert\frac{\partial^2 F}{\partial u_i \partial u_j} (a+v)\right\vert,$$ but I do not feel confident about that. Do you have something else to suggest?
I hope someone could answer both the questions.
Thank you for your time.
For any vector valued function $G$, like here $G=\nabla F$, you can write $$ G(a+u)-G(a)=\int_0^1 G'(a+su)u\,ds $$ Now apply the 1-norm and the triangle inequality for integrals to get $$ \|G(a+u)-G(a)\|_1\le\int_0^1\|G'(a+su)\|_{1,op}\|u\|_1\,ds. $$ $G'$ is here the Hessian matrix of $F$. The operator norm $\|\cdot\|_{1,op}$ is the column-sum norm.
To this one can now apply the mean-value theorem for integration to get $$ \|G(a+u)-G(a)\|_1\le\|G'(a+s^*u)\|_{1,op}\|u\|_1,~~~s^*\in(0,1). $$ One can also write $a+s^*u=z$ with $z\in[a,a+u]$.