Let $F:\mathbb{R}^3\to\mathbb{R}$ be of class $C^\infty$. Let $u\in\mathbb{R}^3$ be nonnull. I am trying to understand if there is a relation between $$|(\nabla\times F)(u)|\quad\text{ and }\quad |\nabla F(u)|,$$ (I mean if we can say if one is greater or less than the other)-
For simplicity, I was looking to the very special case in which $(\nabla\times F)(u) =\frac{\partial F_3}{\partial u_2},$ but I can not how to handle even this easier case.
Could someone please help with that?
Thank you.
It is more or less a consquence of abuse of notation.
We have $$ |\nabla \times F|^2=|\partial_2f_3-\partial_3 f_2|^2+|\partial_3 f_1-\partial_1 f_3|^2+|\partial_1 f_2-\partial_2 f_1|^2 $$ and by absuse of notation, $\nabla$ also denotes the Jacobian $\nabla F$ whose norm is given by $$ |\nabla F|^2=\sum_{i,j=1}^3 |\partial_i f_j|^2. $$ Using the pointwise inequality $|x-y|^2 \leq 2 (|x|^2+|y|^2)$, we deduce that $$ |\partial_i f_j-\partial_jf_i|^2 \leq 2|\partial_i f_j|^2+2|\partial_j f_i|^2 $$ and so $$ |\nabla \times F|^2\leq 2|\nabla F|^2. $$ You could maybe get better inequalities with different norms or using some decomposition argument, but this is the most elementary approach I could think of.