Changing the parameters

Question

I have a constraint equation $ E(x,y) >0 $. I can plot the region satisfying this in $ (x,y) $ plane. I want to change the parameters $(x,y)$ to $ m(x,y) $ and $n(x,y)$, s.t., $m(x,y) $ and $n(x,y)$ are nonlinear in $x$ and $y$, and want plot the region in $(m,n)$ plane.

A typical example of functions $ E(x,y), m(x,y), n(x,y)$:

$ E(x,y) = 7 x^2 Sin(x y) - xy $

$ m(x,y) = (x^2 y +y^3) $

$ n(x,y) = (x^2 + (x\cdot y)^{1/2}) $

Can I do it in Mathematica? I have cooked up the above example. You may consider any convenient case to illustrate the concept.

PS: Mathematica Code

EFun[x_, y_] = 7 x^2  Sin[x y] -  x y


RegionPlot[ EFun[x, y] > 0, {x, 1 E - 4, 100}, {y, 1 E - 4, 100}, PlotPoints -> 4, MaxRecursion -> 4]


m[x_, y_] = x^2 y + y^3


n[x_, y_] = x^2 + (x*y)^(1/2)

Answer 1

EFun[x_, y_] = 7 x^2 Sin[x y] - x y;
m[x_, y_] = x^2 y + y^3;
n[x_, y_] = x^2 + Sqrt[x y];

Invert the relationship between $(x,y)$ and $(m,n)$: this works well for polynomial relationships,

getxy[mm_?NumericQ, nn_?NumericQ] :=
  {x,y} /. Solve[m[x,y]==mm && n[x,y]==nn && x>=0 && y>=0, {x,y}, Reals]

assemble a criterion: not sure if And or Or is required here when several solutions are found for $(x,y)$,

crit[mm_?NumericQ, nn_?NumericQ] := 
  And @@ Thread[EFun @@@ getxy[mm, nn] > 0]

make a region plot:

RegionPlot[crit[mm, nn], {mm, 0, 100}, {nn, 0, 20}]

Alternatively, you can try an analytic inversion:

crit[mm_, nn_] = EFun[x, y] > 0 /. 
  First[Solve[m[x, y] == mm && n[x, y] == nn, {x, y}, Reals]];

This is a bit more tricky though, as it may be difficult to ascertain in general that the first result of Solve is indeed the correct branch of the solutions.