I just want to find an example for $$\dfrac{\partial x}{\partial s} \neq \dfrac{1}{\dfrac{\partial s}{\partial x}}$$
Book and other post like this are all using Jacobian matrices.
They all using the variable $y$, that equation be like $x=f\Big(s(x,y),t(x,y)\Big)$
I'm wonder whether the below way is right, just removed the variable $y$.
let $x=f(s(x),t(x))$.
so we will have $$1 = \frac{dx}{dx} = \frac{\partial x}{\partial x} = \frac{\partial f}{\partial x} $$$$= \frac{\partial f}{\partial s}\cdot \frac{ds}{dx} + \frac{\partial f}{\partial t}\cdot \frac{dt}{dx} $$$$= \frac{\partial f}{\partial s}\cdot \frac{\partial s}{\partial x} + \frac{\partial f}{\partial t}\cdot \frac{\partial t}{\partial x} $$$$= \frac{\partial x}{\partial s}\cdot \frac{\partial s}{\partial x} + \frac{\partial x}{\partial t}\cdot \frac{\partial t}{\partial x}$$
$$1 - \frac{\partial x}{\partial s}\cdot \frac{\partial s}{\partial x} = \frac{\partial x}{\partial t}\cdot \frac{\partial t}{\partial x} $$ So, it should be not always be zero, proved.
Is my proof correct?
