[QUE/TH-01003] TH-PROBLEM

Let
$$\frac{\partial (x,y)}{\partial (a,b)}\,\equiv\,\left|\begin{array}{ll}
\frac{\partial x}{\partial a}&\frac{\partial y}{\partial a}\\
\frac{\partial x}{\partial b}&\frac{\partial y}{\partial b}\\
\end{array}\right|$$

Then show that
$$ \frac{\partial (x,y)}{\partial (a,b)}\frac{\partial (a,b)}{\partial (c,d)}\,=\,\frac{\partial (x,y)}{\partial (c,d)} $$

Remarks : 1. This can be generalised to higher dimensions.

2. This can be found in books - and is very useful in changing variables in multiple integrals.