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

$\int \int_{R_{x y}} f (x, y) d x d y = \int \int_{R_{uv}} f [x (u, v), y (u, v)] \frac{\partial ( x , y )}{\partial ( u , v )} d u d v$

$\frac{\partial ( x , y )}{\partial ( u , v )} = det [\frac{\partial x}{\partial u} \frac{\partial y}{\partial u} \frac{\partial x}{\partial v} \frac{\partial y}{\partial v}]$

This function is called the Jacobian of 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 $(x, y) \to (u, v)$ ; 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

$\frac{\partial ( x , y )}{\partial ( u , v )} = \frac{1}{\frac{\partial ( x , y )}{\partial ( u , v )}}$

Other Learning

Needed to really learn Jacobians because school didn’t teach me well enough.

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

Playlist

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

$J = [\frac{\partial f _{1}}{\partial x} \frac{\partial f _{2}}{\partial x} \frac{\partial f _{1}}{\partial y} \frac{\partial f _{2}}{\partial y}]$

🛠️ Steven Gong

Table of Contents

Change-of-Variable Formula (Jacobian Matrix)

Trick

Other Learning

Graph View

Backlinks