Change-of-Variable Formula (Jacobian Matrix)
The Change-of-Variable Formula
Suppose that the variables and are related to the variables and by the equations , . Then
This function is called the Jacobian of the transformation, and is also denoted by .
Intuitively, the Jacobian is a factor that is introduced to compensate for the distortion of the domain that occurs when we move it from one coordinate system to another.
Restriction to Formula
There is one important restriction on the transformation ; we must not have at any point on the interior of . This condition ensures that the transformation is invertible on the domain of integration.
It can be shown that, as one might hope, that
Needed to really learn Jacobians because school didn’t teach me well enough.
- Khan Academy https://www.youtube.com/playlist?list=PLEZWS2fT1672lJI7FT5OXHJU6cTgkSzV2
For a vector, the jacobian is given by