Ads
related to: non standard calculus exampleskutasoftware.com has been visited by 10K+ users in the past month
Search results
Results from the WOW.Com Content Network
Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval. Example 2: a function f is uniformly continuous on the semi-open ...
Covering nonstandard calculus, it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of nonstandard analysis depend on the existence of the standard part of a finite hyperreal r. The standard part of r, denoted st(r), is a standard real number infinitely close to r ...
Non-standard analysis. Non-standard calculus; Hyperinteger; Hyperreal number; Transfer principle; Overspill; Elementary Calculus: An Infinitesimal Approach; Criticism of non-standard analysis; Standard part function; Set theory. Forcing (mathematics) Boolean-valued model; Kripke semantics. General frame; Predicate logic. First-order logic ...
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model). [1]
The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment.
In non-standard calculus the limit of a function is defined by: = if and only if for all , is infinitesimal whenever x − a is infinitesimal. Here R ∗ {\displaystyle \mathbb {R} ^{*}} are the hyperreal numbers and f* is the natural extension of f to the non-standard real numbers.
In the setting of non-standard calculus, let N be an infinite hyperinteger. The interval [0, 1] has a natural hyperreal extension. The interval [0, 1] has a natural hyperreal extension. Consider its partition into N subintervals of equal infinitesimal length 1/ N , with partition points x i = i / N as i "runs" from 0 to N .
These were used in the initial development of calculus, and are used in synthetic differential geometry. Hyperreal numbers: The numbers used in non-standard analysis. These include infinite and infinitesimal numbers which possess certain properties of the real numbers.
Ads
related to: non standard calculus exampleskutasoftware.com has been visited by 10K+ users in the past month