example if we have cylindrical coordinates : $x = r \ cos(\theta)$ $y = r \ sin(\theta)$ if we treated x as function of $r$ and $\theta$ then:
$dx = \partial x / \partial r\ *\ dr\ + \partial x/ \partial \theta\ * \ d\theta$
$dy = \partial y / \partial r\ *\ dr\ + \partial y/ \partial \theta\ * \ d\theta$
why we cannot use multiplication as $dxdy = (\partial y / \partial r\ *\ dr\ + \partial y/ \partial \theta\ * \ d\theta)*(\partial x / \partial r\ *\ dr\ + \partial x/ \partial \theta\ * \ d\theta)$