Change Of Variable Theorem
Change of Variable Theorem:
Theorem-1 (change of variable formula in the plane). Let S be an elementary region in the xy-plane ( such as disk or parallelogram for example ). Let T : R2 → R2
be an invertible and differentiable mapping and let T (S) be the image of S under T. Then
or more generally
Some notes on the above:
1. We assume T has a inverse function, denoted T-1 . Thus T(x,y) = (u,v) and T-1(u,v) = (x,y).
2. We assume for each (x,y) ∈ S there is one and only one (u,v) that is mapped to, and conversely each (u,v) is mapped to one and only one (x,y).
3. The derivatives of 
and the absolute value of the determinant of the derivatives is
Which implies the area element transforms as;
4. Note that f takes as input x and y but when we change variables our new input are u and v. The map T-1 takes u and v gives x and y and thus we need to evaluate f at 
Remember that we are now integrating over u and v, and thus integrand must be a function of u and v.
5. Note that the formula required an absolute value of determinant. The reason is that the determinant can be negative and we want to see how a small area element transform. Area is supposed to be possitively counted. Note in one variable calculus that 
we need to absolute value to take care of issue such as this.
6. While we started T is a differentiable mapping. our assumption imply T -1 is differential as well.