Editing Epsilon (section)

==Uses==

===International Phonetic Alphabet===
Despite its pronunciation as [[mid<!-- not close-mid, see Arvanti (1999) - Illustrations of the IPA: Modern Greek. --> front unrounded vowel|mid]], in the [[International Phonetic Alphabet]], the Latin epsilon {{IPAc-en|ɛ}} represents [[open-mid front unrounded vowel]], as in the English word ''pet'' {{IPAc-en|p|ɛ|t}}.

===Symbol===
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the [[Latin alphabet|Latin]] letter [[E]]. However, it is commonly used in [[structural mechanics]] with [[Young's Modulus]] equations for calculating tensile, compressive and areal [[Deformation (mechanics)|strain]].

The Greek lowercase epsilon {{code|ε}}, the lunate epsilon symbol {{code|ϵ}}, and the [[Latin epsilon|Latin lowercase epsilon]] {{code|ɛ}} (see above) are used in a variety of places:

* In [[engineering mechanics]], strain calculations ϵ = increase of length / original length. Usually this relates to extensometer testing of metallic materials.
* In [[mathematics]] 
** (In early [[calculus]] or [[nonstandard analysis]]) An infinitesimally small positive quantity is commonly denoted ε.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Epsilon |url=https://mathworld.wolfram.com/Epsilon.html |access-date=2025-01-30 |website=mathworld.wolfram.com |language=en |quote=In mathematics, a small positive infinitesimal quantity, usually denoted {{lang|el|ε}} or {{lang|el|ϵ}}, whose limit is usually taken as {{lang|el|ϵ}}->0.}}</ref> 
*** (In [[Mathematical analysis|analysis]]) By extension, a quantity thought of as "small", "negligible", or, especially, "arbitrarily small", is often denoted ε. For instance, quantities subject to a [[Limit (mathematics)|limit]] which takes them towards zero are often denoted ε; see [[(ε, δ)-definition of limit]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Limit |url=https://mathworld.wolfram.com/Limit.html |access-date=2025-01-30 |website=mathworld.wolfram.com |language=en}}</ref>
** [[David Hilbert|Hilbert]] introduced epsilon terms <math>\epsilon x.\phi</math> as an extension to [[first-order logic]]; see [[epsilon calculus]].
** it is used to represent the [[Levi-Civita symbol]].
** it is used to represent [[dual number]]s: <math>a+b \varepsilon</math>, with <math>\varepsilon^{2}=0</math> and <math>\varepsilon \neq 0</math>.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Dual Number |url=https://mathworld.wolfram.com/DualNumber.html |access-date=2025-01-30 |website=mathworld.wolfram.com |language=en}}</ref>
** it is sometimes used to denote the [[Heaviside step function]].<ref>{{Cite web|url=http://mathworld.wolfram.com/DeltaFunction.html|title=Delta Function|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com |access-date=2019-02-19}}</ref>
** in [[set theory]], the [[epsilon numbers (mathematics)|epsilon numbers]] are [[ordinal number]]s that satisfy the [[Fixed point (mathematics)|fixed point]] ε = ω<sup>ε</sup>. The first epsilon number, ε<sub>0</sub>, is the [[limit ordinal]] of the set {ω, ω<sup>ω</sup>, ω<sup>ω<sup>ω</sup></sup>, ...}.
** in numerical analysis and statistics it is used as the [[error term]]
** in [[group theory]] it is used as the [[idempotent]] group when e is in use as a variable name
* In [[computer science]]
** it often represents the [[empty string]], though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter [[lambda]] (λ).
** the [[machine epsilon]] indicates the upper bound on the relative error due to rounding in floating point arithmetic.<ref>{{Cite book |last=Überhuber |first=Christoph W. |title=Numerical Computation 1: Methods, Software, and Analysis |date=1997 |publisher=Springer |isbn=978-3-540-62058-7 |series=SpringerLink Bücher |location=Berlin, Heidelberg |pages=140 |quote=''eps'' frequently denotes his upper bound on the relative rounding error and is referred to as the ''machine epsilon''.}}</ref>
* In [[physics]], 
** it indicates the [[permittivity]] of a medium; with the subscript 0 (ε<sub>0</sub>) it is the [[permittivity of free space]].<ref>{{Cite web |title=Vacuum electric permittivity |url=https://physics.nist.gov/cgi-bin/cuu/Value?ep0%7Csearch_for=universal_in! |access-date=2025-02-10 |website=physics.nist.gov}}</ref>
** it can also indicate the [[Strain (materials science)|strain]] of a material (a ratio of extensions).<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook|quote= ε linear strain |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
** in [[quantum field theory]], it usually indicates the [[dimensional regularization]] parameter.<ref>{{cite book |last1=Peskin |first1=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction To Quantum Field Theory |date=4 May 2018 |publisher=CRC Press |isbn=978-0-429-97210-2 |edition=2nd |url=https://books.google.com/books?id=HUpaDwAAQBAJ |language=en}}</ref>
* In [[automata theory]], it shows a transition that involves no shifting of an input symbol.
* In [[astronomy]], 
** it stands for the fifth-brightest star in a [[constellation]] (see [[Bayer designation]]).
** Epsilon is the name for the most distant and most visible ring of [[Uranus]].
** In [[planetary science]], ε denotes the [[axial tilt]].<ref>{{Cite book |title=Cyclostratigraphy and astrochronology |date=2018 |publisher=Academic Press, an imprint of Elsevier |isbn=978-0-12-815098-6 |editor-last=Montenari |editor-first=Michael |edition=1st |series=Stratigraphy and Timescales |location=London San Diego, Calif. Cambridge, Mass. Oxford |pages=84 |quote=The Earth's orbital obliquity or axial tilt (ε) is the angle between the Earth's equatorial plane and its orbital plane,}}</ref>
* In [[chemistry]], it represents the [[molar extinction coefficient]] of a [[chromophore]].
* In [[economics]], ε refers to [[Elasticity (economics)|elasticity]].<ref>{{Cite book |last=Free |first=Rhona C. |title=21st century economics: a reference handbook |date=2010 |publisher=Sage |isbn=978-1-4129-6142-4 |location=Thousand Oaks (Calif.) |pages=93-94}}</ref>
* In [[statistics]], 
** it is used to refer to [[Errors and residuals in statistics|error terms]].
** it also can to refer to the degree of [[Mauchly's sphericity test|sphericity]] in [[Repeated measures design|repeated measures ANOVAs]].
* In [[agronomy]], it is used to represent the "[[photosynthetic]] efficiency" of a particular plant or crop.