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November 7th, 2017, 06:02 AM   #1
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Infinitesimal

Is Infinitesimal a number? [The lowest number that exists or epsilon is the lowest number that exist?]

Last edited by skipjack; November 10th, 2017 at 04:27 AM.
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November 7th, 2017, 06:14 AM   #2
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I think you mean "smallest" rather than "lowest".

If $\epsilon$ were the smallest number, what would $\frac\epsilon2$ be?

Infinitesimals don't exist in standard analysis. But this being mathematics, where we set the rules and see what emerges, we can declare the existence of an $\epsilon > 0$ such that $\epsilon < r$ for every positive real number $r$. That results in the Hyperreal numbers which form the basis of (a) non-standard analysis.

Last edited by skipjack; November 10th, 2017 at 04:28 AM.
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November 10th, 2017, 04:12 AM   #3
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Even in non-standard analysis, where infinitesimals are defined, there is NO "smallest" number. There is no smallest infinitesimal.

Last edited by skipjack; November 10th, 2017 at 04:29 AM.
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November 10th, 2017, 04:24 AM   #4
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Infinitesimals find more use in Engineering and Physics, where they are very useful, than in Mathematics, where they are frowned on.
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November 10th, 2017, 04:41 AM   #5
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Originally Posted by studiot View Post
Infinitesimals find more use in Engineering and Physics, where they are very useful, than in Mathematics, where they are frowned on.
Are they used for anything more than simplifying the theory of calculus? Or are you confusing them with differentials?
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November 10th, 2017, 06:00 AM   #6
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Quote:
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Are they used for anything more than simplifying the theory of calculus? Or are you confusing them with differentials?
Derive the equation of motion for the simple pendulum.
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November 10th, 2017, 06:06 AM   #7
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Originally Posted by v8archie View Post
Are they used for anything more than simplifying the theory of calculus? Or are you confusing them with differentials?
Yeah, they are used in perturbation theory. For example, a density, concentration or temperature perturbation in a fluid element leads to convective mass transport as a natural consequence of adding an infinitesimal change into the equations related to the equation of state of the medium.
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November 10th, 2017, 10:14 AM   #8
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I thought perturbation theory used small changes and approximations rather than trying to claim exact results via infinitesimals.
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November 10th, 2017, 10:39 AM   #9
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I thought perturbation theory used small changes and approximations rather than trying to claim exact results via infinitesimals.
You are right. But we find infinitesimals to be so useful some Physicists forget the distinction. This attitude leads to what I call "Physics Math" which is merely a loose application of Mathematical techniques. A better example of using infinitesimals in Physics is in topics such as the "principle of virtual work." This leads (more or less) to Lagrangian Mechanics.

-Dan
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Last edited by topsquark; November 10th, 2017 at 10:43 AM.
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November 10th, 2017, 10:57 AM   #10
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... we find infinitesimals to be so useful some Physicists forget the distinction.
No problem with that (as long as they are used properly). The results have been proved to be accurate in all applications.
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