## 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 : R^{2 }→ R^{2}

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.