Plotting conformal mappings: $(x, y) \mapsto (x^2, y)$

Question

I'm currently studying transformation geometry, and I've been trying to figure out a good way to plot conformal mappings on $\mathbb{R}^2$ of the form $$ \mathbf{f} : \pmatrix{x \\ y} \mapsto \pmatrix{f_1(x) \\ f_2(y)} , $$ for example with $f_1(x) = x^2$ and $f_2(y) = y$. All the examples I've seen so far (e.g. here and here) are in the complex plane, with a single valued function, and seem to be complex analysis stuff.

For context, I was inspired by the animations in 3Blue1Brown's videos. However, in addition to linear transformations like $$ R = \pmatrix{ \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) } , $$ I'm also interested in seeing the effect of nonlinear functions on the gridlines of the Cartesian plane.

Answer 1

Also from complex analysis, just by using Re and Im to obtain f1[x,y] and f2[x,y]

    f1[x_, y_] = x^2 - y^2;
    f2[x_, y_] = 2 x*y;
    ParametricPlot[{f1[x, y], f2[x, y]}, {x, -4, 4}, {y, -4, 4}, 
     MeshFunctions -> Automatic, Mesh -> 8, 
     MeshShading -> {{LightRed, LightGreen}, {LightBlue, LightYellow}}, 
     Axes -> False, PlotRange -> All, BoundaryStyle -> None, 
     FrameLabel -> {u, v}, LabelStyle -> {FontFamily -> "Times", Blue}, 
     PlotPoints -> 50]

Black and White

f1[x_, y_] = x^2 - y^2;
f2[x_, y_] = 2 x*y;
ParametricPlot[{f1[x, y], f2[x, y]}, {x, -4, 4}, {y, -4, 4}, 
 MeshFunctions -> Automatic, Mesh -> 8, 
 MeshStyle -> Directive[Thickness[0.015], Cyan], 
 MeshShading -> {{Black, White}, {White, Black}}, Axes -> False, 
 PlotRange -> All, BoundaryStyle -> Red, FrameLabel -> {u, v}, 
 LabelStyle -> {FontFamily -> "Times", Blue}, PlotPoints -> 50]