Clear["Global`*"];
a = .3*2 π;
η = x*Cos[a] + y*Sin[a];
ζ = -x*Sin[a] + y*Cos[a];
g = Show[
ContourPlot[η, {x, 0, 10}, {y, 0, 10}, Contours -> {1, 2, 3},
ContourShading -> None],
ContourPlot[ζ, {x, 0, 10}, {y, 0, 10},
Contours -> Range[-3, 3], ContourShading -> None],
PlotRange -> All, AspectRatio -> Automatic];
{g, g /. {x_Real, y_Real} :> {Sqrt[x], Sqrt[y]}}
a = .3*2 π;
Show[ContourPlot[
x*Cos[a] + y*Sin[a] /. {x -> x^2, y -> y^2} // Evaluate, {x, 0,
10}, {y, 0, 10}, Contours -> {1, 2, 3}, ContourShading -> None],
ContourPlot[-x*Sin[a] + y*Cos[a] /. {x -> x^2, y -> y^2} //
Evaluate, {x, 0, 10}, {y, 0, 10},
Contours -> {-3, -2, -1, 0, 1, 2, 3}, ContourShading -> None]]
a = .3*2 π;
η = x*Cos[a] + y*Sin[a];
ζ = -x*Sin[a] + y*Cos[a];
ParametricPlot[{Sqrt[x], Sqrt[y]}, {x,
0, 10}, {y, 0, 10},
MeshFunctions -> {Function[{x1, y1, x, y}, η],
Function[{x1, y1, x, y}, ζ]},
Mesh -> {{1, 2, 3}, {-3, -2, -1, 0, 1, 2, 3}}, PlotPoints -> 150,
MaxRecursion -> 6, PlotStyle -> None, Frame -> False,
BoundaryStyle -> None]
