I was given a function $F(x,y,z)=(z^c,x^c,y^c)$ and asked to find divergence and curl. My initial answer was $0$, but i don’t think that’s right.
I noticed the brackets weren’t the typical $\langle,\rangle$ vector field notation. Vector field $F$ should be $\langle z^c/r, x^c/r, y^c/r\rangle$, where r=magnitude of $x,y,z$. That is what I should be finding the divergence and curl of, correct? My answer is non-trivial (zero) in that case.
Best Answer
Let's do a special case: $F(x,y,z)=(z,x,y)$ i.e. $c=1$. Then $$\operatorname{div}(F)=\partial_x(z)+\partial_y(x)+\partial_z(y)=0.$$ The curl is given by $$\operatorname{curl}(F)=\nabla \times F=\left(\partial_y (y)-\partial_z(x), \partial_z(z)-\partial_x(y), \partial_x(x)-\partial_y(z) \right)=(1,1,1)\ne 0.$$ Here, the $\partial_x$ denotes (for example) the partial derivative with respect to $x$.