Hodge Dual - Derivatives in Three Dimensions

Derivatives in Three Dimensions

The combination of the operator and the exterior derivative d generates the classical operators grad, curl, and div, in three dimensions. This works out as follows: d can take a 0-form (function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (applied to a 3-form it just gives zero). For a 0-form, the first case written out in components is identifiable as the grad operator:

The second case followed by is an operator on 1-forms that in components is the operator:

Applying the Hodge star gives:

The final case prefaced and followed by, takes a 1-form to a 0-form (function); written out in components it is the divergence operator:

One advantage of this expression is that the identity, which is true in all cases, sums up two others, namely that and . In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star.

One can also obtain the Laplacian. Using the information above and the fact that then for a 0-form, :