pic=ParametricPlot3D[{r*Cos[theta],r*Sin[theta],r^2},{r,0,Sqrt[2]},{theta,0,2*Pi},BoxRatios->{2*Sqrt[2],2*Sqrt[2],2},Boxed->False,Mesh->20,AxesLabel->{"X","Y","Z"},ViewPoint->{1.3,-2.4,1}]

Export["elliptic_paraboloid_circular_domain.png",pic,ImageSize->300]