# Changing the parameters

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)


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.