So, I never took any PDE in my math undergrad and somehow I got shoved into an applied PDE course.

I was to prove that the Laplace EQ is invariant under rotation.

So, i'm pretty good at making concrete examples and then seeing what mechanisms are happening and making a general proof.

I proved that Uxx + Uyy = Ux'x' + Uy'y' where x'y' arise from a fixed rotation of theta. Essentially I can prove the statement for two dimensions...How I do this generally I do not understand.

I've seen proofs online that start with v(x) : = u*Ox, and y = Ox. My question are x and y vectors or just variables?

if they are vectors I get a column vector y = Orthogonal matrix * column vectors of x....what do I need to do next?