An infinitesimal is either a positive infinitesimal, a negative infinitesimal, or zero. In $\mathbb {R}$ there is only one infinitesimal, zero - this is precisely the Archimedean property of $\mathbb {R}$.

Dec 4, 2016 · Covering 1.4 of Keisler's Elementary Calculus, "Slope and Velocity; The Hyperreal Line" That chapter defines: A number $\\epsilon$ is said to be infinitely small, infinitesimal, if: $-a < \\epsil...

Aug 3, 2017 · There is an $\epsilon$ (infinitesimal) thrown in there as well. How do you understand these extremely small values and what do I need to do to account for them when calculating very …

Jan 7, 2016 · If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students? What I am asking for are specific techniques for explaining infinitesimal...

Aug 17, 2016 · By far the most direct way to talk about "infinitely short line segments" is to use nonstandard analysis. In standard mathematics, there are various ways to make sense of …

Dec 31, 2014 · For the definition of Infinitesimal, wikipedia says In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or, so small t...

5 When considering an infinitesimal distance/interval/in calculus, what is the intuitive interpretation? Is it too small to be measurable but still has some distance on an unattainable scale? Are there different …

Dec 4, 2024 · Your point about how the infinitesimal diffeomorphism can be used to fix a gauge for linearized gravity is a good one. That's more likely to be what the professor is doing than what I wrote.

Dec 23, 2024 · Different methods apply depending on what your goal is, and some approaches involve working with models of Smooth Infinitesimal Analysis containing both nilpotent infinitesimals and …

Apr 24, 2013 · and define an infinitesimal number as the difference between a convergent geometric series and its sum: $ x+1 -\displaystyle\sum_ {i=0}^ {n\rightarrow\infty} \left (\frac {x} {x+1}\right)^i$ If …