12:33 AM
$dx^{j}$ is a tangent vector
$dx^{j}$ can capture an infinitesimal change in a function value, thus bypasses the non uniqueness problem of infintesimals as traditionally outlined by Lebniz
Now applying explosive generalisation, we can consider the following mathematical object:
$M(dx)$ is a mathematical object with the property being infinitesimal or capture an infinitesimal property, meaning:
$M(dx) = \{x \in M | \phi(x)\}$
where $\phi$ reads "is related to or has the property of being infintesimal"
Now recall the definition of an infinitesimal as an element $\epsilon$ in a nonarchimedian ring $R$ such that for any $x \in R$, $x\epsilon < 1$
hmm... need to figure out how to formalise $M(dx)$... perhaps the literature already have some examples
