- Hodge Star

Now, we define the Hodge star operation for a 1-form *α* as

$$(\star \mathfrak{a})\_o = \mathfrak{g}^\sharp(\mathfrak{a})\_\downarrow \text{Area}\_\mathcal{O} \tag{76}$$

with respect to an orientation *o* of the plane. The star operator maps a 1-form to a twisted 1-form. Naturally, we can define a Hodge star operation for a twisted 1-form *α*ˇ as:

$$
\star \check{\mathfrak{a}} = \mathfrak{g}^{\sharp}(\check{\mathfrak{a}}\_{\mathcal{O}}) \lrcorner \text{Area}\_{\mathcal{O}}.\tag{77}
$$

Then, the Hodge operator maps a twisted 1-form to an untwisted 1-form. The multiple operations of are shown in Figure 33. In this figure, we can see

$$
\star \star = -\mathrm{Id},
\tag{78}
$$

where Id is the identity operator. Therefore, defines the complex structure in the two-dimensional plane. By using the Hodge star, we can represent Ohm's law with a scalar sheet conductance *G* as *K* = *G E*. Note that the Hodge operator can be defined for other *p*-forms, but the sign of generally depends on the order *p*, the dimension of the space, and the metric signature, rather than Equation (78) [59].

**Figure 33.** Complex structure of Hodge operations for untwisted and twisted 1-forms.

#### *4.7. Summary of Basic Equations in Differential-Form Approach*

Now, we summarize the basic equations with differential forms for a two-dimensional resistive sheet with a nonuniform scalar conductance. The electric field is represented by a 1-form *E*, while the current density field is given by a twisted 1-form *K*. KVL and KCL are formulated as

$$\text{d}E = 0,\tag{79}$$

$$\text{d}K = 0,\tag{80}$$

respectively. The scalar Ohm's law is rewritten as

$$\mathcal{K} = G \star E \tag{81}$$

with a scalar sheet conductance *G*(= 1/*R*). These equations are schematically shown in Figure 34. Although we only focused on Cartesian coordinates, Equations (79)–(81) are coordinate free. Therefore, we can use an arbitrary coordinate for analysis. Another feature of the differential-form formalism is the exclusion of the metric in Equations (79) and (80). The metric appears through the Hodge star in Equation (81). Thus, Equations (79) and (80) are metric-free equations and easy to be discretized while keeping the geometrical structure, as we see later.

**Figure 34.** Structure of basic equations in a two-dimensional resistive sheet.

#### *4.8. Keller–Dykhne Duality with Differential Forms*

Now, we formulate Keller–Dykhne duality with differential forms. Electric and current fields are represented by untwisted and twisted 1-forms, respectively. To exchange these fields with two different kinds of orientations, we need to fix an orientation of the plane. Here, we define a twisted scalar Ω = {(Ω)*o*|*<sup>o</sup>* <sup>=</sup>, } satisfying (Ω) = +1 and (Ω)− = −1. The pseudoscalar Ω is regarded as the plane orientation . For a twisted form *ω*<sup>ˇ</sup> , Ω extracts the component Ω*ω*ˇ = *ω*<sup>ˇ</sup> . Now, we consider the replacement as

$$E^{\star} = \mathcal{R}\_{n\ell} \Omega^{\oplus} K\_{\prime} \tag{82}$$

$$K^\star = -G\_{\text{tot}} \Omega^\odot E \tag{83}$$

with respect to a reference resistance *<sup>R</sup>*ref(= 1/*G*ref). Clearly, these fields satisfy

$$\text{d}E^\* = 0,\tag{84}$$

$$\mathbf{d}\mathcal{K}^\* = 0.\tag{85}$$

From Equation (81) with Equations (82) and (83), we obtain

$$
\mathbb{K}^\star = G^\star \star E^\star,\tag{86}
$$

where we use Equation (78) and *G* = (*<sup>G</sup>*ref)2/*<sup>G</sup>* for a scalar *G*.
