Editing Scalar field (section)

== Definition ==
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>{{cite book |first=Tom |last=Apostol |author-link=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]].

[[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.]]

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.