I know that it is not an objective question, but I ask for advices and tips for succesfully compute expressions like :
$\nabla \cdot \nabla f$
$ \nabla\cdot \nabla \times F$
$ \nabla\cdot(fF)$
$ \nabla\times\nabla \times F$
$\nabla\times (fF)$
I'm sorry if my question is vague... It's just that I have many troubles to compute those things. I'm not even sure if what I wrote is something that makes sense ! And perhaps, I'm not the only one struggling with those ugly symbols ?
Could you please help me ? I really need to know how to develop expressions with those operators, and even after a few months trying, I still do not have a good method.
Say you have a vector field $F=\langle F_1,F_2,F_3\rangle.$ Then it's divergence is $$\nabla\cdot F=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$$ On the other hand, if you have a scalar function $f(x,y,z)$ then it's gradient is $$\nabla f=\langle f_x,f_y,f_z\rangle,$$ where $f_x$ is the partial derivative of $f$ with respect to $x$ and so on. Now putting these two things together you $$\text{div grad f }=\nabla \cdot (\nabla f)=\nabla\cdot \langle f_x,f_y,f_z\rangle = f_{xx}+f_{yy}+f_{zz}.$$
For the last one, consider $F=\langle A,B,C\rangle.$ Then $$\text{ rot (rot F)} = \nabla \times \langle C_y-B_z,A_z-C_x,B_x-A_y \rangle$$ $$ =\langle (B_x-A_y)_y-(A_z-C_x)_z,(C_y-B_z)_z-(B_x-A_y)_x,(A_z-C_x)_x-(C_y-B_z)_y\rangle$$ $$ = \langle B_{xy}-A_{yy}-A_{zz}+C_{xz},C_{yz}-B_{zz}-B_xx+A_{yx},A_{zx}-C_{xx}-C_{yy}+BB_{zy} \rangle.$$ You need to work from the inner operator to the outer one when solving problems of this kind.