Assume we have four functions $\in C^\infty$ from $\mathbb R^2 $ to $\mathbb R$, written as follows: $$ x=x(u,v) , \quad u=u(r,s) $$ $$ y=y(u,v) , \quad v=v(r,s) $$ then, assuming the composition $x(u(r,s),v(r,s))$ is properly defined, $$ \frac{\partial x}{\partial r} = \frac{\partial x}{\partial u} \frac{\partial u}{\partial r} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial r}. $$
But, in the particular case where $(r,s)=(x,y)$, we have that $\frac{\partial x}{\partial x} =1 $, but the right hand side gives us $$ \frac{\partial x}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial x} $$ which equals 2?
What am I getting wrong here? Does my misunderstanding follows from a notation error?
Thanks
$\frac{\partial x}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial x}$ doesn't equal $2$. This is one of those instances where regarding differential notation as cancellable fractions leads you astray.