Find a function f such that gradient of f =F if F(x,y,z)=?

Answer 1

I tried to find f by integrating the partial derivatives, but since 1/r is multiplying the whole vector, I just took it out, I'm not sure if I can do that. Like this:

∂f∂x(x,y,z)=x

∂f∂y(x,y,z)=y

∂f∂z(x,y,z)=z

thus

f(x,y,z)=x22+g(y,z)

f(x,y,z)=y22+h(x,z)

f(x,y,z)=z22+k(x,y)

for some functions g, h, and k, so if g=y22+z22, h=x22+z22 and k=x22+y22, the function f is:

f(x,y,z)=1r(x22+y22+z22)=12r⋅r2=r2

Am I correct? If not, how can I solve this correctly, should I integrate x/r, y/r and z/r instead?