# Field

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In PDE, a **field** refers to any function (or section) on the underlying spatial or spacetime domain. Common examples of fields include

*Real scalar fields*, which take values in the real line . Examples include the field*u*appearing in NLW or KdV.*Complex scalar fields*, which take values in the complex plane . Examples include the field*u*appearing in NLS, or the field appearing in MKG.*Vector fields*, which take place in a vector space, or perhaps in some vector bundle over the manifold (such as the tangent manifold).*Connections*, which look superficially similar to vector fields but transform differently under gauge transformations. Examples include the field appearing in MKG or YM.*Tensor fields*, which are sections of some tensor product of a base vector bundle. Examples include the metric*g*appearing in the Einstein equations*Spinor fields*, which are sections of a spinor bundle, and which appear primarily in Dirac equations.

This notion of a field arises from physics and is unrelated to the algebraic concept of a field (i.e. a commutative division ring).