It is clear that the Lorentz gauge-fixing term ([[partial derivative].sub.[mu]][A.sup.[mu]]) is a 0-form which emerges out from the 1-form ([A.sup.(1)] = d[x.sup.[mu]][A.sub.[mu]]) due to application of the coexterior derivative ([delta] = - * d*) which reduces the degree of a form by one.
We quote here the result of operation of [??] on [[??].sup.(1)] as 0-form; namely,
In other words, we have super Hodge duality * on the 0-form as follows:
For a 0-form, barycentric coordinates themselves constitute the suitable interpolant, that is
where [[lambda].sub.i](u) is simply the value of [[lambda].sub.i] at the point u and [[alpha].sup.(0)] is an arbitrary 0-form. From the properties of barycentric coordinates, it is trivial to verify that this interpolation recovers the nodal values (contractions), that is, [[alpha].sup.(p)] (u = [[sigma].sub.0,I]) = [[alpha].sup.(0).sub.i].
In (18), we used the fact that for a 0-form the domain of integration [[sigma].sub.0,I] in (9) is simply a point (node), and hence the integration (contraction) reduces to the evaluation of a function (0-form) at such point.
Because of this, in the 3-d case for example, 0-forms (evaluated at a point) exhibit full continuity (associated to scalar potential fields), 1-forms exhibit tangential continuity (associated to "intensity" vector fields), 2-forms exhibit surface normal continuity (associated to "flux" vector fields), and 3-forms may exhibit stepwise discontinuities (e.g., scalar charge densities).
Now [Mathematical Expression Omitted] is a closed 0-form, i.e., a constant.
The term [Mathematical Expression Omitted] is a d-closed 0-form, i.e., a constant.