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).