Standard part function: Difference between revisions

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==Applications==

All the traditional notions of calculus are expressed in terms of the standard part function, as follows.

===Derivative===

The standard part function is used to define the derivative of a function ''f''. If ''f'' is a real function, and ''h'' is infinitesimal, and if ''f''′(''x'') exists, then

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

===Integral===

Similarly, givenGiven 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].

===Limit===

Given a sequence <math>(u_n)</math>, its limit is defined by <math>\lim_{n\to\infty}u_n=\text{st}(u_H)</math> where <math>H\in{}^\ast\mathbb{N}\setminus\mathbb{N}</math> is an infinite index. Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.

==See also==