# Change-of-Variable Formula (Jacobian Matrix)

The Change-of-Variable Formula

Suppose that the variables $x$ and $y$ are related to the variables $u$ and $v$ by the equations $x=x(u,v)$, $y=y(u,v)$. Then

$โซโซ_{R_{xy}}f(x,y)dxdy=โซโซ_{R_{uv}}f[x(u,v),y(u,v)]โโ(u,v)โ(x,y)โโdudv$

$โ(u,v)โ(x,y)โ=det[โuโxโโuโyโโโvโxโโvโyโโ]$

This function is called the

Jacobianof the transformation, and is also denoted by $J$.

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 $(x,y)โ(u,v)$;

we must not have $J=0$ at any point on the interior of $R_{uv}$. This condition ensures that the transformation is invertible on the domain of integration.

### Trick

It can be shown that, as one might hope, that

$โ(u,v)โ(x,y)โ=โ(u,v)โ(x,y)โ1โ$

### Other Learning

Needed to really learn Jacobians because school didnโt teach me well enough.

Learning SLAM through Cyrill Stachniss, and he uses the Jacobian to explain the Structure Matrix.

Playlist

- Khan Academy https://www.youtube.com/playlist?list=PLEZWS2fT1672lJI7FT5OXHJU6cTgkSzV2
- https://www.youtube.com/watch?v=wCZ1VEmVjVo&ab_channel=Mathemaniac

What 3Blue1Brown was explaining is that all a Jacobian is zooming into the function graph and you then see things as Linear Transformation. Local Linearity.

For a $[f_{1}(x,y)f_{2}(x,y)โ]$ vector, the jacobian is given by

$J=[โxโf_{1}โโxโf_{2}โโโyโf_{1}โโyโf_{2}โโ]$