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| [[File:Scalar_field.png|thumb|right|A scalar field such as temperature or pressure, where intensity of the field is graphically represented by different hues of color.]]
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| In [[mathematics]] and [[physics]], a '''scalar field''' associates a [[scalar (mathematics)|scalar]] value to every point in a space. The scalar may either be a [[scalar (mathematics)|mathematical number]], or a [[scalar (physics)|physical quantity]]. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space (or spacetime). Examples used in physics include the [[temperature]] distribution throughout space, the [[pressure]] distribution in a fluid, and spin-zero quantum fields, such as the [[Higgs field]]. These fields are the subject of [[scalar field theory]].
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| == Definition ==
| | To one's offense, you might contain Gunboats which can simply shoot at enemy safeguarding coming from a very long range and Landing Products which you must load when you train units for example Rifleman, Heavy, Zooka, Warrior and Fish tanks. For more information about [http://prometeu.net clash of clans cheats iphone] look into our own page. To your village defenses, you might have definitely structures like Mortar, Bike Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Skyrocket Launcher to assist your family eradicate enemies.<br><br>Resorting from band blueprint that will help your besprinkle blueprint gives some sort of special authentic picture. This accumbent time arbor is usually scaled evenly. But it's adamantine to prove to be able to acquaint what's activity now within the bottom-left bend now. The ethics are subsequently bunched up you will certainly not acquaint them afar nowadays.<br><br>Pay attention to a game's evaluation when purchasing a present. This evaluation will allow you to comprehend what age level clash of clans hack tool is most suitable for and will reveal to you when the sport is undoubtedly violent. It can help you figure out whether you should buy the sport.<br><br>In Clash of Clans Secrets and cheats (a brilliant popular ethnic architecture and arresting fearless by Supercell) participants may well acceleration up accomplishments for instance building, advance or exercise routine troops with gems which can be bought for absolute profit. They're basically monetizing the accurate player's impatience. Each single amusing architecture daring My spouse and i apperceive of manages to do it.<br><br>Actual not only provides undertake tools, there is perhaps Clash of Clans hack into no survey by individual. Strict anti ban system make users to utilize shed pounds and play without an hindrance. If internet poker players are interested in receiving the program, they are definitely required to visit great site and obtain the most important hack tool trainer immediately. The name of the internet business is Amazing Cheats. A number of web-sites have different types created by software by which men or women can get past a difficult situation stages in the fixture.<br><br>Be careful about letting your student play online video games, especially games with feed sound. There could be foul language in these kind channels, in addition to a lot of bullying behavior. Could also be child predators in these products chat rooms. Know what your child is buying and surveil these chitchat times due to his or her own protection.<br><br>Now that you have read this composition, you need to a good easier time locating as well as , loving video games in your own life. Notwithstanding your favored platform, from your cellphone to any own computer, playing as well enjoying video gaming allow you to take the benefit of the worries of the [http://imageshack.us/photos/actual+busy actual busy] week get additional information. |
| Mathematically, a scalar field on a [[Region_(mathematical_analysis)|region]] ''U'' is a [[real-valued function|real]] or [[complex-valued function]] or [[distribution (mathematics)|distribution]] on ''U''.<ref>{{citation|first=Tom|last=Apostol|authorlink=Tom Apostol|title=Calculus, Volume II|publisher=Wiley|year=1969|edition=2nd}}</ref><ref>{{springer|title=Scalar|id=s/s083240}}</ref> The region ''U'' may be a set in some [[Euclidean space]], [[Minkowski space]], or more generally a subset of a [[manifold]], and it is typical in mathematics to impose further conditions on the field, such that it be [[continuous function|continuous]] or often [[continuously differentiable]] to some order. A scalar field is a [[tensor field]] of order zero,<ref>{{springer|id=s/s083260|title=Scalar field}}</ref> and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, [[density bundle|density]], or [[differential form]].
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| [[File:Scalar Field.ogv|thumb|The scalar field of <math>\sin (2\pi(xy+\sigma))</math> oscillating as <math>\sigma</math> increases. Red represents positive values, purple represents negative values, and sky blue represents values close to zero.]]
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| Physically, a scalar field is additionally distinguished by having [[units of measurement]] associated with it. In this context, a scalar field should also be independent of the coordinate system used to describe the physical system—that is, any two [[observer (special relativity)|observer]]s using the same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as [[vector field]]s, which associate a [[Euclidean vector|vector]] to every point of a region, as well as [[tensor field]]s and [[spinor|spinor fields]].{{Citation needed|date=June 2012}} More subtly, scalar fields are often contrasted with [[pseudoscalar]] fields.
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| ==Uses in physics==
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| In physics, scalar fields often describe the [[potential energy]] associated with a particular [[force]]. The force is a [[vector field]], which can be obtained as the [[gradient]] of the potential energy scalar field. Examples include:
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| *[[Potential field]]s, such as the Newtonian [[gravitational potential]], or the [[electric potential]] in [[electrostatics]], are scalar fields which describe the more familiar forces.{{Citation needed|date=June 2012}}
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| * A [[temperature]], [[humidity]] or [[pressure]] field, such as those used in [[meteorology]].
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| ===Examples in quantum theory and relativity===
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| * In [[quantum field theory]], a [[Bosonic field|scalar field]] is associated with spin-0 particles. The scalar field may be real or complex valued. Complex scalar fields represent charged particles. These include the charged [[Higgs field]] of the [[Standard Model]], as well as the charged [[pions]] mediating the [[strong nuclear interaction]].<ref>Technically, pions are actually examples of [[pseudoscalar meson]]s, which fail to be invariant under spatial inversion, but are otherwise invariant under Lorentz transformations.</ref>
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| * In the [[Standard Model]] of elementary particles, a scalar [[Higgs field]] is used to give the [[lepton]]s and [[W and Z bosons|massive vector bosons]] their mass, via a combination of the [[Yukawa interaction]] and the [[spontaneous symmetry breaking]]. This mechanism is known as the [[Higgs mechanism]].<ref>{{cite journal|author=P.W. Higgs|journal=Phys. Rev. Lett|volume=13|issue=16|pages=508|month=Oct.|title=Broken Symmetries and the Masses of Gauge Bosons|doi=10.1103/PhysRevLett.13.508|year=1964|bibcode = 1964PhRvL..13..508H }}</ref> A candidate for the [[Higgs boson]] was first detected at CERN in 2012.
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| * In [[scalar theories of gravitation]] scalar fields are used to describe the gravitational field.
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| * [[Scalar-tensor theory|scalar-tensor theories]] represent the gravitational interaction through both a tensor and a scalar. Such attempts are for example the [[Pascual Jordan|Jordan]] theory <ref>P. Jordan ''Schwerkraft und Weltall'', Vieweg (Braunschweig) 1955.</ref> as a generalization of the [[Kaluza-Klein theory]] and the [[Brans-Dicke theory]].<ref>C. Brans and R. Dicke; ''Phys. Rev. '''124'''(3): 925'', 1961.</ref>
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| :* Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the [[Standard Model]].<ref>A. Zee; ''Phys. Rev. Lett. '''42'''(7): 417'', 1979.</ref><ref>H. Dehnen ''et al.''; Int. J. of Theor. Phys. '''31'''(1): 109'', 1992.</ref> This field interacts gravitationally and [[Yukawa interaction|Yukawa]]-like (short-ranged) with the particles that get mass through it.<ref>H. Dehnen and H. Frommmert, ''Int. J. of theor. Phys. '''30'''(7): 987'', 1991.</ref>
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| * Scalar fields are found within superstring theories as [[dilaton]] fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor.<ref>C.H. Brans; "The Roots of scalar-tensor theory", arXiv:gr-qc/0506063v1, June 2005.</ref>
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| * Scalar fields are supposed to cause the accelerated expansion of the universe (inflation <ref>A. Guth; ''Phys. Rev. '''D23''': 347'', 1981.</ref>), helping to solve the [[horizon problem]] and giving an hypothetical reason for the non-vanishing [[cosmological constant]] of cosmology. Massless (i.e. long-ranged) scalar fields in this context are known as [[inflaton]]s. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields.<ref>J.L. Cervantes-Cota and H. Dehnen; ''Phys. Rev. '''D51''', 395'', 1995.</ref>
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| ==Other kinds of fields==
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| *[[Vector field]]s, which associate a [[vector (geometry)|vector]] to every point in space. Some examples of [[vector field]]s include the [[electromagnetic field]] and the Newtonian gravitational field.
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| *[[Tensor field]]s, which associate a [[tensor]] to every point in space. For example, in [[general relativity]] gravitation is associated with the tensor field called [[Einstein tensor]]. In [[Kaluza-Klein theory]], spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary [[dimension|four-dimensional]] gravitation plus an extra set, which is equivalent to [[Maxwell's equations]] for the [[electromagnetic field]], plus an extra scalar field known as the "[[dilaton]]".{{Citation needed|date=June 2012}} The dilaton scalar is also found among the massless bosonic fields in [[string theory]].
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| == See also ==
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| * [[Scalar field theory]]
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| * [[Vector-valued function]]
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:Scalar Field}}
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| [[Category:Quantum field theory]]
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| [[Category:Multivariable calculus]]
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