-Stokes' Theorem

How may one relate an exterior derivative to the boundary of a chain? Green's theorem

$$\oint\_{\partial \mathcal{S}} (f \text{dx} + g \text{dy}) = \iint\_{\mathcal{S}} \left( \frac{\partial g}{\partial \mathbf{x}} - \frac{\partial f}{\partial y} \right) \text{d}x \text{dy} \tag{66}$$

can be rewritten as the following equation for a 2-chain *S* with a 1-form *α*:

$$
\int\_{S} \mathbf{d}a = \int\_{\partial S} a.\tag{67}
$$

Equation (67) generally holds for higher-dimensional spaces, and it is known as Stokes' theorem. Stokes' theorem relates the boundary operator *∂* with the exterior derivative d.

## - Twisted 1-Form

Next, we consider a representation of current density. As discussed in Section 2.3, current flow should be calculated for an outer-oriented 1-chain, so the current density is an outer-oriented 1-cochain. The outer-oriented 1-chain *c*ˇ was represented as *c*ˇ = {*c*<sup>ˇ</sup>*o*|*<sup>o</sup>* <sup>=</sup>, } by using the inner-oriented 1-chain *c*ˇ*o* depending on a plane orientation *o*. Thus, current density *K* should be represented as *K* = {*Ko*|*o* <sup>=</sup>, } with the two 1-forms satisfying

$$K\_{-\mathcal{O}} = -K\_{\mathcal{O}} \tag{68}$$

where −*o* represents the opposite orientation of *o*. The set of two 1-forms satisfying Equation (68) is called a *twisted* 1-form. On the other hand, ordinary forms are called *untwisted*. A twisted 1-form *α*ˇ at a point *P* is an outer-oriented covector *α*ˇ *P*, which is depicted as the inner-oriented lines in Figure 31a. To stress the twist, we put the check symbol (ˇ) for twisted objects, but sometimes omit the mark to reduce the notation complexity. In Figure 31b, the twisted 1-form *K* is depicted by (local) stream lines. We can count the total current flow across a curve by the integration. The integration of *K* along an outer-oriented 1-chain *c*ˇ is defined as

$$\int\_{\mathbb{Z}} \mathbf{K} := \int\_{\mathbb{Z}\_{\sigma}} \mathbf{K}\_{\sigma}. \tag{69}$$

The exterior derivative of *K* is also defined as

$$(\mathbf{d}\mathbf{K})\_o = \mathbf{d}K\_o.\tag{70}$$

KCL states that

$$\int\_{\partial \mathcal{S}} K = \int\_{\mathcal{S}} \mathbf{d}K = 0 \tag{71}$$

for any outer-oriented 2-cell *S* ˇ . Considering small *S* ˇ , we obtain d*K* = 0 as KCL.

#### - Metric Tensor

To express Ohm's law, we will define the Hodge star operation. To define the Hodge star operation, we need to introduce a metric tensor. In the two-dimensional Euclidean space E2, we can define the inner product of vectors *u* = *<sup>u</sup>xex* + *<sup>u</sup>yey* and *v* = *<sup>v</sup>xex* + *<sup>v</sup>yey* as

$$\mathcal{S}(\mathfrak{u}, \mathfrak{v}) = \mathfrak{u} \cdot \mathfrak{v} \tag{72}$$

with *u* · *v* = *uxvx* + *<sup>u</sup>yvy*. Here, *g* is called the *metric tensor*. We can define a covector *g*(*u*) for a vector *u* satisfying

$$\mathcal{g}^{\flat}(\mathfrak{u}) = \mathcal{g}(\mathfrak{u}, \sqcup). \tag{73}$$

Therefore, *v g*(*u*) = *g*(*<sup>u</sup>*, *v*) holds. The operation of *g* on *ex* and *ey* is graphically shown in Figure 32. The inverse map of *g* is written as *g* = (*g*)−1.

$$\mathbf{g^{\flat}(\longrightarrow\_{\mathbf{e}\_{\times}})} = \left| \begin{array}{c} \uparrow \\ \downarrow \\ \downarrow \end{array} \right| \qquad \mathbf{g^{\flat}(\left[ \begin{array}{c} \bullet \\ \end{array} \right])} = \begin{array}{c} \overline{\xrightarrow{\mathbf{e}\_{\times} \cdot \mathbf{e}\_{\times}}} \end{array}$$

**Figure 32.** Conversion between a vector and a covector.

#### - Area Form as a Twisted 2-Form

With respect to the metric of E2, {*<sup>e</sup>x*, *<sup>e</sup>y*} is an orthogonal basis: *g*(*<sup>e</sup>i*, *ej*) = *δij* (*δij*: Kronecker delta). Using the orthogonal coordinate, we can define a twisted 2-form "Area" as

$$\mathbf{Area}\_{\triangleleft} = \mathbf{dx} \wedge \mathbf{dy},\tag{74}$$

$$\text{Area}\_{\triangle} = -\text{dx} \wedge \text{dy}.\tag{75}$$

The area form measures the *unsigned* area of an outer-oriented 2-chain.
