Editing Exterior algebra (section)

=== Electromagnetic field ===
In [[Theory of relativity|Einstein's theories of relativity]], the [[electromagnetic field]] is generally given as a [[differential form|differential 2-form]] <math> F = dA </math> in [[4-space]] or as the equivalent [[Antisymmetric tensor|alternating tensor field]] <math> F_{ij} = A_{[i,j]} = A_{[i;j]}, </math> the [[electromagnetic tensor]].  Then <math> dF = ddA = 0 </math> or the equivalent Bianchi identity <math> F_{[ij,k]} = F_{[ij;k]} = 0. </math>
None of this requires a metric.

Adding the [[Lorentz metric]] and an [[Orientability#Orientability of differentiable manifolds|orientation]] provides the [[Hodge star operator]] <math> \star </math> and thus makes it possible to define <math> J = {\star}d{\star}F </math> or the equivalent tensor [[divergence]] <math> J^i = F^{ij}_{,j} = F^{ij}_{;j} </math> where <math> F^{ij} = g^{ik}g^{jl}F_{kl}. </math>