TITLE: Differential form vs Hyperreal vs Dual number
QUESTION [6 upvotes]: Differential forms, hyperreals, and dual numbers all seem to sort of do something similar: formalize the notion of the infinitesimal. 
How are they related to each other, and in what ways are they different? I know that the hyperreals sort of "extend" the real numbers via nonstandard analysis, but how is this different from what dual numbers do? Similarly, differential forms seem to be able to generalization to higher dimensions -- is this something that hyperreals cannot do?

REPLY [4 votes]: I think the notation can causes confusion:
Differential forms are not infinitesimal quantities although the notation (e.g. dx) may incorrectly suggest otherwise. The d in front of dx stands for the exterior derivative and should not be confused with the d in front of an infinitesimal, e.g. dX. As a form dx is a linear function that takes a vector as argument and spits out a finite real number.