# Change Of Variable Theorem

**Special Cases:**

**Theorem** (change of variable theorem: Polar Coordinates). Let x = r cosθ, y = r sinθ the inverse function are:

Let D be an elementary region in the xy-plane and let D* be the corresponding region in the rθ plane. Then

For example if D is the region x^{2} y^{2} ≤ 1 in the xy-plane then D* is the rectangle [0,1] * [0,2π] in the rθ plane.

**Theorem** ( change of Variable Theorem: Cylindrical Coordinates). Let with and z arbitrary; note the inverse function are:

Let D be a elementary region in the xyz-plane and let D* be the coresponding region in the rθz - plane. Then

**Theorem** ( Change of Variable Theorem : Spherical Coordinates) Let with

Note that the angle φ is the angle made with z - axis; many books interchanges the role of φand θ. Let D be the elementary region in the xyz-plane and let D* be the coresponding region in the ρθφ-plane. Then

Note the most common mistake is to have incorrect bounds of integration. Hence this is the basic Special **Cases of Change of Variables Theorem.**