# Maxwell's equations

• ... the orginal field equations explicitly contain the magnetic vector potential, ${\displaystyle {\overrightarrow {A}}}$ ... In Maxwell's original formulaton, Faraday's ${\displaystyle {\overrightarrow {A}}}$ field was central and had physical meaning. The magnetic vector potential was not arbitrary, as defined by boundary conditions and choice of gauge as we will discuss; they were said to be gauge invariant. The original equations are thus often called the Faraday-Maxwell theory.
• ... applications in gauge field theories and the physics of condensed matter. The starting point here is now quite well known: expressing the Maxwell equations for an electromagnetic field over Lorentz space as the Euler-Lagrange equations for a Lagrangian defined on the connections on a ${\displaystyle U(1)}$ bundle, where the electromagnetic potential becomes the connection and the field tensor its curvature. The freedom of choice of "gauge" for the potential is a fundamental fact which stems, in the geometrical picture, from the lack of a preferred trivialisation of the bundle.