B.2.2 For the Vector Function

The Neumann boundary integral for $\vec{H}_1$ from (5.75) is written in the form

$$\begin{split}\int_{\mathcal{C}_{N2}}\lambda_i\vec{n}\cdot\lef... ...\mathrm{d}s = & = \left[D\right]\left\{c\right\} \end{split}$$ (B.34)

$$D_{ij}^e = \mu\int_{\mathcal{C}^e_k}\lambda_i^e\vec{n}_k\cdot\vec{N}_j^e \mathrm{d}s, i\in[1;3], j\in[1;3], k\in[1;3].$$ (B.35)

Along Edge $12$ :

$$\begin{split}D_{11}^e & = \mu\int_{\mathcal{C}^e_1}\lambda_1^... ...\vec{r}_{31} - \vec{r}_{12}\cdot\vec{r}_{23}\right) \end{split}$$ (B.36)

$$\begin{split}D_{12}^e & = \mu\int_{\mathcal{C}^e_1}\lambda_1^... ...{\mu{}l_{23}}{12F_e} \vec{r}_{12}\cdot\vec{r}_{12} \end{split}$$ (B.37)

$$\begin{split}D_{13}^e & = \mu\int_{\mathcal{C}^e_1}\lambda_1^... ...c{\mu{}l_{31}}{6F_e} \vec{r}_{12}\cdot\vec{r}_{12} \end{split}$$ (B.38)

$$D_{21}^e= -\frac{\mu{}l_{12}}{12F_e}\left(\vec{r}_{12}\cdot\vec{r}_{31} - 2 \vec{r}_{12}\cdot\vec{r}_{23}\right)$$ (B.39)

$$D_{22}^e= -\frac{\mu{}l_{23}}{6F_e} \vec{r}_{12}\cdot\vec{r}_{12}$$ (B.40)

$$D_{23}^e= \frac{\mu{}l_{31}}{12F_e} \vec{r}_{12}\cdot\vec{r}_{12}$$ (B.41)

$$D_{3j}^e= \mu\int_{\mathcal{C}^e_1}\lambda_3^e \vec{n}_3\cdot\vec{N}_j^e \mathrm{d}s = 0, \lambda_3^e = 0 \mathrm{on Edge} 12.$$ (B.42)

Along Edge $23$ :

$$D_{1j}^e= \mu\int_{\mathcal{C}^e_2}\lambda_1^e \vec{n}_1\cdot\vec{N}_j^e \mathrm{d}s = 0, \lambda_1^e = 0 \mathrm{on Edge} 23$$ (B.43)

$$\begin{split}D_{21}^e & = \mu\int_{\mathcal{C}^e_2}\lambda_2^... ...c{\mu{}l_{12}}{6F_e} \vec{r}_{23}\cdot\vec{r}_{23} \end{split}$$ (B.44)

$$\begin{split}D_{22}^e & = \mu\int_{\mathcal{C}^e_2}\lambda_2^... ...\vec{r}_{12} - \vec{r}_{23}\cdot\vec{r}_{31}\right) \end{split}$$ (B.45)

$$\begin{split}D_{23}^e & = \mu\int_{\mathcal{C}^e_2}\lambda_2^... ...{\mu{}l_{31}}{12F_e} \vec{r}_{23}\cdot\vec{r}_{23} \end{split}$$ (B.46)

$$D_{31}^e = \frac{\mu{}l_{12}}{12F_e} \vec{r}_{23}\cdot\vec{r}_{23}$$ (B.47)

$$D_{32}^e = -\frac{\mu{}l_{23}}{12F_e}\left(\vec{r}_{23}\cdot\vec{r}_{12} - 2 \vec{r}_{23}\cdot\vec{r}_{31}\right)$$ (B.48)

$$D_{33}^e = -\frac{\mu{}l_{31}}{6F_e} \vec{r}_{23}\cdot\vec{r}_{23}.$$ (B.49)

Along Edge 31:

$$\begin{split}D_{11}^e & = \mu\int_{\mathcal{C}^e_3}\lambda_1^... ...c{\mu{}l_{12}}{6F_e} \vec{r}_{31}\cdot\vec{r}_{31} \end{split}$$ (B.50)

$$\begin{split}D_{12}^e & = \mu\int_{\mathcal{C}^e_3}\lambda_1^... ...{\mu{}l_{23}}{12F_e} \vec{r}_{31}\cdot\vec{r}_{31} \end{split}$$ (B.51)

$$\begin{split}D_{13}^e & = \mu\int_{\mathcal{C}^e_3}\lambda_1^... ...c{r}_{23} - 2 \vec{r}_{31}\cdot\vec{r}_{12}\right) \end{split}$$ (B.52)

$$D_{2j}^e= \mu\int_{\mathcal{C}^e_3}\lambda_2^e \vec{n}_2\cdot\vec{N}_j^e \mathrm{d}s = 0, \lambda_2^e = 0 \mathrm{on Edge} 31$$ (B.53)

$$D_{31}^e = -\frac{\mu{}l_{12}}{12F_e} \vec{r}_{31}\cdot\vec{r}_{31}$$ (B.54)

$$D_{32}^e = \frac{\mu{}l_{23}}{6F_e} \vec{r}_{31}\cdot\vec{r}_{31}$$ (B.55)

$$D_{33}^e = -\frac{\mu{}l_{31}}{12F_e}\left(2 \vec{r}_{31}\cdot\vec{r}_{23} - \vec{r}_{31}\cdot\vec{r}_{12}\right).$$ (B.56)