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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?

The Art Of Repetition
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1110101001
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  • You may be interested in looking at some texts on synthetic differential geometry – Willie Wong Mar 20 '17 at 02:58
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    One difference of hyperreals vs dual numbers is the hyperreals still form a field, but the dual numbers can't always have division (specifically the element $0 + b\epsilon$ has no inverse). Also, see here for more discussion. – Mark Schultz-Wu Mar 20 '17 at 03:03
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    Another difference of all of the above is that hyperreals/dual numbers are sort of number systems (elements of them are "constant"), and differential forms are certain kinds of functions (elements of them are not as constant). Additionally, while $d^2\omega = 0$ for differential forms, and $\epsilon^2 = 0$ for dual numbers, $d^2\omega = d\circ d(\omega)$ is applying the differential operator twice, while $\epsilon\times \epsilon$ is multiplication. – Mark Schultz-Wu Mar 20 '17 at 03:05

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The dual numbers are useful both in physics and mathematics, including developing the theory of Lie algebras. In fields like analysis and differential geometry, the dual numbers are of limited utility because they don't have a number of properties possessed by more advanced number systems such as the hyperreals.

The dual numbers cannot be included in the hyperreals because of the peculiar properties of the dual numbers that make them useful in physics, e.g., $\epsilon^2=0$ which cannot be satisfied in the hyperreals since the latter form a field.

When Riemann originally developed a field today called Riemannian geometry, he defined a metric as a weighted sum of quadratic expressions in the differentials $dx^i$ where each $dx^i$ is an infinitesimal increment of the coordinate $x^i$.

The expressions of the form $dx^i$ were eventually reinterpreted in an infinitesimal-frei manner, as sections of the cotangent bundle on the manifold. This is a more advanced topic than infinitesimal calculus.

Mikhail Katz
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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.

Hans Westman
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