Standard part function: Difference between revisions

Revision as of 00:49, 17 December 2020

Revision as of 22:52, 26 December 2020

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Nonstandard analysis deals primarily with the pair <math>\mathbb{R}\subseteq{}^{\ast}\mathbb{R}</math>, where the [[hyperreal number|hyperreal]]s <math>{}^{\ast}\mathbb{R}</math> are an [[ordered field]] extension of the reals <math>\mathbb{R}</math>, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a [[monad (nonstandard analysis)|monad]], or '''halo''') of hyperreals infinitely close to it. The standard part function associates to a [[Wikt:finite|finite]] [[hyperreal number|hyperreal]] ''x'', the unique standard real number ''x₀'' that is infinitely close to it. The relationship is expressed symbolically by writing

:<math>\,\mathrmoperatorname{st}(x)=x_0.</math>

The standard part of any [[infinitesimal]] is 0. Thus if ''N'' is an infinite [[hypernatural]], then 1/''N'' is infinitesimal, and st(1/''N'')&nbsp;=&nbsp;0.

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:<math>f'(x) = \operatorname{st}\left(\frac {f(x+h)-f(x)}h\right).</math>

Alternatively, if <math>y=f(x)</math>, one takes an infinitesimal increment <math>\Delta x</math>, and computes the corresponding <math>\Delta y=f(x+\Delta x)-f(x)</math>. One forms the ratio <math>\frac{\Delta y}{\Delta x}</math>. The derivative is then defined as the standard part of the ratio:

:<math>\frac{dy}{dx}=\mathrmoperatorname{st}\left( \frac{\Delta y}{\Delta x} \right)</math>.

===Integral===

Given a function <math>f</math> on <math>[a,b]</math>, one defines the integral <math>\int_a^b f(x)dx</math> as the standard part of an infinite Riemann sum <math>S(f,a,b,\Delta x)</math> when the value of <math>\Delta x</math> is taken to be infinitesimal, exploiting a [[hyperfinite set|hyperfinite]] partition of the interval [a,b].