Field

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 $\mathbb {R}$. Examples include the field u appearing in NLW or KdV.
• Complex scalar fields, which take values in the complex plane $\mathbb {C}$. Examples include the field u appearing in NLS, or the field $\phi$ 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 $A_{\alpha }$ 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).