Levi-Civita field Information & Levi-Civita field Links at HealthHaven.com

The Levi-Civita field is a non-Archimedean field, i.e., a system of numbers containing infinite and infinitesimal quantities. Its members can be constructed as formal series of the form

$\sum_{q\in\mathbb{Q}} a_q\epsilon^q \qquad ,$

where the $a q$ are real, $\mathbb{Q}$ is the set of rational numbers, and ε is to be interpreted as a positive infinitesimal. The reals are embedded as series in which all coefficients vanish except for $a 0$ .

The nonvanishing coefficients $a q$ must be a left-finite set, i.e., for any member of the set, there must be finitely many members less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to dictionary ordering of the list of coefficients, which is equivalent to the assumption that ε is an infinitesimal.

[edit] Examples

7ε is an infinitesimal that is greater than ε, but less than every positive real number.
ε² is less than ε, and is also less than rε for any real r.
1+ε differs infinitesimally from 1.
ε^1/2 is greater than ε, but still less than every positive real number.
1/ε is greater than any real number.
$1+\epsilon+\frac{1}{2}\epsilon^2+\ldots+\frac{1}{n!}\epsilon^n+\ldots$ is interpreted as e^ε.
$1+\epsilon+2\epsilon^2+\ldots+n!\epsilon^n+\ldots$ is a valid member of the field, because the series is to be construed formally, without any consideration of convergence.

[edit] Extentions and applications

The field can be algebraically closed by adjoining an imaginary unit, or by letting the coefficients be complex. It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating point. It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.^[1]

[edit] References

^ Khodr Shamseddine, "Analysis on the Levi-Civita Field: A Brief Overview", http://www.uwec.edu/surepam/media/RS-Overview.pdf

[edit] External links

A web-based calculator for Levi-Civita numbers

[0] Khodr Shamseddine, "Analysis on the Levi-Civita Field: A Brief Overview", http://www.uwec.edu/surepam/media/RS-Overview.pdf

[1]

Contents

[edit] Examples

[edit] Extentions and applications

[edit] References

[edit] External links