Chapter 6.3, Problem 46E

(For those who have studied multivariable calculus.) Let T be an invertible linear transformation from ${ℝ}^2$ to ${ℝ}^2$ , represented by the matrix M. Let ${\Omega }_1$ be the unit square in ${ℝ}^2$ and ${\Omega }_2$ its image under T. Consider a continuous function $f\left(x,y\right)$ from ${ℝ}^2$ to $ℝ$ , and define the function $g\left(u,v\right)=f\left(T\left(u,v\right)\right)$ . What is the relationship between the following two double integrals?
${\displaystyle \underset{{\Omega }_2}{\iint }f\left(x,y\right)dA}$ and ${\displaystyle \underset{{\Omega }_1}{\iint }g\left(u,v\right)dA}$
Your answer will involve the matrix M. Hint: What happens when $f\left(x,y\right)=1$ , for all x, y?