# Jacobian and chain rule

Let $u$, $v$ be differentiable functions of $x$, $y$ and $x$, $y$ be differentiable functions of $s$, $t$.  Then the connection

 $\displaystyle\frac{\partial(u,v)}{\partial(s,t)}\;=\;\frac{\partial(u,v)}{% \partial(x,y)}\cdot\frac{\partial(x,y)}{\partial(s,t)}$ (1)

between the Jacobian determinants is in .

Proof.  Starting from the right hand side of (1), where one can multiply the determinants (http://planetmath.org/Determinant2) similarly as the corresponding matrices (http://planetmath.org/MatrixMultiplication), we have

 $\left|\begin{matrix}u_{x}&u_{y}\\ v_{x}&v_{y}\\ \end{matrix}\right|\cdot\left|\begin{matrix}x_{s}&x_{t}\\ y_{s}&y_{t}\\ \end{matrix}\right|\;=\;\left|\begin{matrix}u_{x}x_{s}+u_{y}y_{s}&u_{x}x_{t}+u% _{y}y_{t}\\ v_{x}x_{s}+v_{y}y_{s}&v_{x}x_{t}+v_{y}y_{t}\\ \end{matrix}\right|\;=\;\left|\begin{matrix}u_{s}&u_{t}\\ v_{s}&v_{t}\\ \end{matrix}\right|.$

Here, the last stage has been written according to the general chain rule (http://planetmath.org/ChainRuleSeveralVariables).  But thus we have arrived at the left hand side of the equation (1), which hereby has been proved.

Remark.  The rule (1) is only a visualisation of the more general one concerning the case of functions of $n$ variables.

Title Jacobian and chain rule JacobianAndChainRule 2013-03-22 18:59:45 2013-03-22 18:59:45 pahio (2872) pahio (2872) 4 pahio (2872) Example msc 15-00 msc 26B05 msc 26B10