Jacobi Matrix and Multiple Intgrals

[Kork]

Homework Statement

Let D be the set of points (x,y) in R^2 for which 0 is ≤ x ≤ 1 and 0 ≤ y ≤ 1. Find a function g: R^2 --> R for which:

∫_0^1 ∫_0^1 h(x,y)dxdy = ∫_0^1∫_0^1 h(y^5, x^3) * g(x,y)dxdy

is true for all functions h: D--> R integrable over D

In the question before this I was asked to find a jacobi matrix and determinant for f(x,y) = (y^5,x^3)

I found that the determinant is -15y^4x^3

Homework Equations

The Attempt at a Solution

∬_D f(x,y)dxdy= ∬_S g(v^5,u^3) * -15y^4x^2

Is this my final result? If not, can I get some help on how to go on?

 

[LCKurtz]

Homework Statement

Let D be the set of points (x,y) in R^2 for which 0 is ≤ x ≤ 1 and 0 ≤ y ≤ 1. Find a function g: R^2 --> R for which:

∫_0^1 ∫_0^1 h(x,y)dxdy = ∫_0^1∫_0^1 h(y^5, x^3) * g(x,y)dxdy

is true for all functions h: D--> R integrable over D

In the question before this I was asked to find a jacobi matrix and determinant for f(x,y) = (y^5,x^3)

I found that the determinant is -15y^4x^3

Homework Equations

The Attempt at a Solution

∬_D f(x,y)dxdy= ∬_S g(v^5,u^3) * -15y^4x^2

Is this my final result? If not, can I get some help on how to go on?

Your answer shouldn't have 4 variables and needs the integration variables.

Your problem certainly suggests the change of variables $x=u^5,\ y=v^3$. You need to check that maps the square to the square. Then use your change of variables theorem to express your integral in terms of $u$ and $v$. Your result should have just $u$ and $v$ variables in it. Of course, they are dummy variables in the final result.