I am just having a bit of trouble understanding what I am being asked.
The ie: in title says $\phi [x(u,v), y(u,v)]$
Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write it in terms of $\nabla^2_{u,v} \phi (u,v)$ and $d_z \, f(z)$. Part b) is showing that if one is harmonic the other is harmonic too, seem simple once I get going.
I know that this is just applying the chain rule to $\nabla^2_{x,y} \phi (x,y)$ with $u$ and $v$ plugged in but I don't quite understand how $f(z) = u(x,y) + i v(x,y)$ enters the picture.
What I got so far is $\nabla^2_{x,y} \phi (x,y) = \frac{\partial^2 \phi[(x(u,v),y(u,v)]}{\partial x^2} + \frac{\partial^2 \phi[(x(u,v),y(u,v)]}{\partial y^2}$ but I am not sure what to do from here or if this is the right interpretation since, again, I don't see how $f(z)$ comes in.