Editing Field equation (section)

{{Short description|Partial differential equation describing physical fields}}
In [[theoretical physics]] and [[applied mathematics]], a '''field equation''' is a [[partial differential equation]] which determines the dynamics of a [[physical field]], specifically the time evolution and spatial distribution of the field. The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled equations which must be solved simultaneously. Field equations are not [[ordinary differential equation]]s since a field depends on space and time, which requires at least two variables.

Whereas the "[[wave equation]]", the "[[diffusion equation]]", and the "[[continuity equation]]" all have standard forms (and various special cases or generalizations), there is no single, special equation referred to as "the field equation".

The topic broadly splits into equations of [[classical field theory]] and [[quantum field theory]]. Classical field equations describe many physical properties like temperature of a substance, velocity of a fluid, stresses in an elastic material, electric and magnetic fields from a current, etc.<ref>{{cite book |last1=Fetter|first1=A. L.|last2=Walecka|first2=J. D.| title=Theoretical Mechanics of Particles and Continua|year=1980|isbn= 978-0-486-43261-8|publisher=Dover|pages=439, 471}}</ref> They also describe the fundamental forces of nature, like electromagnetism and gravity.<ref>{{cite book|first=J. D.|last=Jackson|author-link=John David Jackson (physicist)|title=Classical Electrodynamics|edition=2nd|year=1975|orig-year=1962|isbn=0-471-43132-X|publisher=[[John Wiley & Sons]]|page=[https://archive.org/details/classicalelectro00jack_0/page/218 218]|url-access=registration|url=https://archive.org/details/classicalelectro00jack_0/page/218}}</ref><ref>{{cite book|last1=Landau|first1=L.D.|author-link1=Lev Landau|last2=Lifshitz|first2=E.M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=[[Butterworth&ndash;Heinemann]]|isbn=0-7506-2768-9|year=2002|orig-year=1939|page=297}}</ref> In quantum field theory, particles or systems of "particles" like [[electron]]s and [[photon]]s are associated with fields, allowing for infinite degrees of freedom (unlike finite degrees of freedom in particle mechanics) and variable particle numbers which can be [[Matter creation|created]] or [[Annihilation|annihilated]].