Are infinitesimals real
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Many fields of physics, notably the theory of elasticity and the macroscopic electrodynamics, operate with a physically small volume - that is the quantities in these theories are understood as averages over volumes containing $N_A$ number of atoms/molecules - not truly infinitesimal, but which ate treated as infinitesimally small in mathematical equations.Some people think that you can informally sneak them in and out of the real numbers without causing confusion.
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These numbers, which can be defined precisely, behave like very large natural numbers. Infinitesimals play as big a role in properly-done everyday real analysis as ghosts do in everyday lifenamely, not a very big role. He used the hyperreals (R), an extension of real numbers to include infinitesimal numbers and infinite numbers. Still, infinitesimals have their mathematical role: e.g., discrete spectra can be calculated and analyzed using continuous Schrodinger equation. Although infinitesimals had been around for some time (albeit barred from modern calculus), Robinson gave them a precise definition. On a microscopic level not everything is continuous - at least not all the time.Given a function f of a real variable x and an interval a, b of the real line. Every function has a unique slope such that for all nilsquare infinitesimals . Integrals theory Integrals are the sum of infinite summands, infinitely small.
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The fact that every function is linear is formalized by the Kock-Lawvere axiom: Axiom. The real line contains a set of infinitesimal numbers. They are relevant to physics as much as the fields and other mathematical objects in physical theories can be considered continuous, and therefore described in terms of differential equations. In Smooth Infinitesimal Analysis, the set of real numbers is replaced with the real line. As has correctly pointed out, infinitesimals are a mathematical concept.