Jacobians

#Math

Topics

$$

J = \left\lvert \frac{ \partial (x,y) }{ \partial (u,v) } \right\rvert =\begin{bmatrix}
\frac{ \partial x }{ \partial u } & \frac{ \partial x }{ \partial v } \\\frac{ \partial y }{ \partial u } & \frac{ \partial y }{ \partial v }
\end{bmatrix}
$$

• Two variable version

$$

J = \left\lvert \frac{ \partial (x,y,z) }{ \partial (u,v,w) } \right\rvert =\begin{bmatrix}
\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{bmatrix}
$$

• Three variable version

Examples

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

• See spherical coordinates

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

• See polar coordinates and cylindrical coordinates