Suppose $x=x(t)$. Let $dx$ and $dt$ be 1-forms.
Does there exist a rigorous theorem for converting $dx=f(t)dt$ into the derivative ("fraction") expression $\frac{dx}{dt}=f(t)$?
Is this possible in general for other possibly multivariate functions (or does it require that $x$ is only a function of $t$). Or is this just false?
My guess is that by the linear independence of the basis differential form dt, it is possible to expand in the following manner and simply equate coefficients of $dt$.
$dx=\frac{dx}{dt}dt$
Therefore, $\frac{dx}{dt}=f(t)$.