How do I solve (∂/∂x+∂/∂y+∂/∂z)^2 of u

Question

I'm trying to solve a question in which it is given that u=f(x,y,z) and it is asked to find (∂/∂x+∂/∂y+∂/∂z)^2 of u . How do I solve this? Should I apply the formula (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca ? Or any other easy method ?

Answer 1

This is merely a suggestion on how to book keep your calculation. I will leave the actual calculations to you.

I suggest that you do it in two steps. First, calculate $$ h=\Bigl(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\Bigr)u=\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}. $$ (Note, that you by symmetry only need to do one of the differentiations.) After simplification, I get $$ h=\frac{3}{x+y+z}. $$ Then, calculate $$ \Bigl(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\Bigr)^2u=\Bigl(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\Bigr)h=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial y}+\frac{\partial h}{\partial z}. $$ When I do the calculation, I get the result $$ -\frac{9}{(x+y+z)^2}. $$