Jacobian

$$ \begin{align} J=\frac{\partial (u,v,w)}{\partial (x,y,z)}\\ J= \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{vmatrix} \end{align} $$

Property #

JJ’=1 where J’=$\frac{\partial (x,y,z)}{\partial (u,v,w)}$

Proof #

$$ \begin{align} JJ’ = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{vmatrix}= \begin{vmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{vmatrix} =1 \end{align} $$

Change of Variables using the Jacobian #

When transforming (x,y)→(new X, new Y)=$(f_x(x,y),f_y(x,y))$, there’s a transformation that’s doesn’t have to be linear overall but linear locally represented by the matrix:

$$ \begin{bmatrix} \frac{\partial f_x}{\partial x} & \frac{\partial f_x}{\partial y}\\ \frac{\partial f_y}{\partial x} & \frac{\partial f_y}{\partial y} \end{bmatrix} $$

The determinant of the matrix represents how much the area is dilated during the transformation: $J=\frac{\partial (f_x,f_y)}{\partial (x,y)}$

Therefore, (the infinitesimally small area in the initial (x,y)) should only be ($\frac{1}{J}\cdot$ the infinitesimally small area in the final $(f_x,f_y)$)

$$ \begin{align} dx\cdot dy=\frac{1}{J}\cdot df_x \cdot df_y\\ dx\cdot dy=\frac{\partial (x,y)}{\partial (f_x,f_y)}\cdot df_x\cdot df_y \end{align} $$

Example #

Relation between cartesian system’s infinitesimal area to polar system’s infinitesimal area

$$ \begin{align} dx\cdot dy=\frac{\partial (x,y)}{\partial (r,\theta)}\cdot dr\cdot d\theta\\ =\begin{vmatrix} \cos\theta & -r\sin \theta\\ \sin\theta & r\cos \theta\\ \end{vmatrix} dr\cdot d\theta \\ dx\cdot dy = rdr\cdot d\theta \end{align} $$