Say I want to integrate a nice enough function $f: \mathbb{R^2}\rightarrow\mathbb{R}$ over a region $R \subset\mathbb{R^2},$ hence evaluating
$$ \int_R f(x,y) \;\mathrm{dxdy}. $$
But since this is too hard, I perform the integration in coordinates $(u, v)$ different to $(x, y).$ Say the region $R$ in $(u,v)$ is $R'.$
Using $x = x(u,v)$ and $y=y(u,v)$, I obtain $$ \mathrm{dx} = \frac{\partial x}{\partial u} \mathrm{du}+ \frac{\partial x}{\partial v}\mathrm{dv} \quad\quad \mathrm{dy}=\frac{\partial y}{\partial u} \mathrm{du} + \frac{\partial y}{\partial v} \mathrm{dv}. $$ So I thought I just 'multiply' them (using distributive property) to get $$ \mathrm{dxdy} = \left( \frac{\partial x}{\partial u} \mathrm{du}+ \frac{\partial x}{\partial v}\mathrm{dv}\right) \left(\frac{\partial y}{\partial u} \mathrm{du} + \frac{\partial y}{\partial v} \mathrm{dv}\right) = \\ \frac{\partial x}{\partial u}\mathrm{du}\frac{\partial y}{\partial u} \mathrm{du}+ \frac{\partial x}{\partial v}\mathrm{dv}\frac{\partial y}{\partial u} \mathrm{du}+ \frac{\partial x}{\partial u}\mathrm{du}\frac{\partial y}{\partial v} \mathrm{dv}+ \frac{\partial x}{\partial v}\mathrm{dv}\frac{\partial y}{\partial v} \mathrm{dv}. $$ I know I want something proportional to $\mathrm{dudv}$ so I ignore the $\mathrm{du}\mathrm{du}$ and $\mathrm{dv}\mathrm{dv}$ terms, this leaves
$$ \mathrm{dxdy} = \frac{\partial x}{\partial v}\mathrm{dv}\frac{\partial y}{\partial u} \mathrm{du}+ \frac{\partial x}{\partial u}\mathrm{du}\frac{\partial y}{\partial v} \mathrm{dv} $$ Given the context in which these differentials were motivated to me and the fact I know the solution to this integration problem, I went ahead and assumed (kind of) linearity so factor out the partial derivatives, such $$ \mathrm{dxdy} = \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\mathrm{dv} \mathrm{du}+ \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}\mathrm{du} \mathrm{dv}. $$ Now I'm stuck. I know I need $\mathrm{dvdu} = -\mathrm{dudv}.$ But why does that not work? I'm so close to the determinant of the Jacobian. Why can't I just say
$$ \int_R f(x,y) \;\mathrm{dxdy} = \int_{R'} f(x(u,v),y(u,v))\;\left(\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\mathrm{dv} \mathrm{du}+ \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}\mathrm{du} \mathrm{dv}\right)= \\ \int_{R'} f(x(u,v),y(u,v))\;\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\mathrm{dv} \mathrm{du}+\int_{R'} f(x(u,v),y(u,v))\; \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}\mathrm{du} \mathrm{dv}. $$ Finally use Fubini to integrate what ever I want first. - It's as if differentials have a direction.