This notebook plots z = u(r, θ), with polar coordinates, on the disk r ≤ 1, 0 ≤ θ < 2π.
x = r cos(θ)
y = r sin(θ)

u[r_, θ_] := (r Cos[θ])^2 - (r Sin[θ])^2  (* this is polar version of u = x^2 - y^2 *)
ParametricPlot3D[{r Cos[θ], r Sin[θ], u[r, θ]}, {r, 0, 1}, {θ, 0, 2 π}, BoundaryStyle -> Black, BoxRatios ->{1,1,.7}]

u[r_, θ_] := r^2 Cos[2θ] (* another polar version of u = x^2 - y^2 *)
ParametricPlot3D[{r Cos[θ], r Sin[θ], u[r, θ]}, {r, 0, 1}, {θ, 0, 2 π}, BoundaryStyle -> Black, BoxRatios ->{1,1,.7}]