Jacobians

Topics

$$J=\left|\frac{\partial (x,y)}{\partial (u,v)}\right|=\left[\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right]$$

• Two variable version

$$J=\left|\frac{\partial (x,y,z)}{\partial (u,v,w)}\right|=\left[\begin{array}{ccc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}& \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}& \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u}& \frac{\partial z}{\partial v}& \frac{\partial z}{\partial w}\end{array}\right]$$

• Three variable version

Examples

$J_{\text{spherical}}={\rho }^2\sin \varphi$

• See spherical coordinates

$J_{\text{polar}}=J_{\text{cylindrical}}=r$

• See polar coordinates and cylindrical coordinates