B.2 Surface normal and surface vector quantities

The components of the unit surface normal $\hat{n}$ are

$$S^{-1} \left [ r r_{\phi} \sin{\phi} - r r_{\theta} \sin{\theta} \cos{\theta} \cos{\phi} + r^2 \sin^2{\theta} \cos{\phi} \right ]$$ n_x =

$$S^{-1} \left [-r r_{\phi} \cos{\phi} - r r_{\theta} \sin{\theta} \cos{\theta} \sin{\phi} + r^2 \sin^2{\theta} \sin{\phi} \right ]$$ n_y =

$$S^{-1} \left [r r_{\theta} \sin^2{\theta} + r^2 \cos{\theta} \sin{\theta} \right ]$$ n_z = (33)

and the derivatives of the surface vector $\vec{X}$ are

$$\frac{\partial \vec{X}}{\partial \phi} \left (r_{\phi}\sin{\theta}\cos{\phi} - r \sin{\theta}\sin{\phi},... ...n{\theta}\sin{\phi} + r \sin{\theta}\cos{\phi},\: r_{\phi}\cos{\theta} \right )$$ = (34)

$$\frac{\partial \vec{X}}{\partial \theta} \left (r \cos{\theta}\cos{\phi} + r_{\theta}\sin{\theta}\cos{\phi... ...eta}\sin{\theta}\sin{\phi},\: r_{\theta}\cos{\theta} - r \sin{\theta} \right ).$$ = (35)

The derivatives of the components of $\hat{n}$ with respect to $\phi$ are given by the following formula, where i = x, y,z:

$$\frac{\partial n_i}{\partial \phi} = S^{-1} \left [ a_i - b_i \left ( \frac{r_{\phi}}{r} + \frac{c_i}{S^2} \right ) \right ]$$ (36)

where the functions a_i, b_i, and c_i are:

$$r^2_{\phi} \sin{\phi} + r r_{\phi \phi} \sin{\phi} + r r_{\phi} \... ...s{\theta} \cos{\phi} - r r_{\theta \phi} \sin{\theta} \cos{\theta} \cos{\phi} +$$ a₁ =

$$r r_{\theta} \sin{\theta} \cos{\theta} \sin{\phi} + 2 r r_{\phi} \sin^2{\theta} \cos{\phi} - r^2 \sin^2{\theta} \sin{\phi}$$

$$r r_{\phi} \sin{\phi} - r r_{\theta} \sin{\theta} \cos{\theta} \cos{\phi} + r^2 \sin^2{\theta} \cos{\phi}$$ b₁ =

$$r^2 (r_{\phi} r_{\phi \phi} + r_{\theta} r_{\theta \phi} \sin^2{\theta} +r r_{\phi} \sin^2{\theta}$$ c₁ =

$$- r^2_{\phi} \cos{\phi} - r r_{\phi \phi} \cos{\phi} + r r_{\phi}... ...s{\theta} \sin{\phi} - r r_{\theta \phi} \sin{\theta} \cos{\theta} \sin{\phi} -$$ a₂ =

$$r r_{\theta} \sin{\theta} \cos{\theta} \cos{\phi} + 2 r r_{\phi} \sin^2{\theta} \sin{\phi} + r^2 \sin^2{\theta} \cos{\phi}$$

$$- r r_{\phi} \cos{\phi} - r r_{\theta} \sin{\theta} \cos{\theta} \sin{\phi} + r^2 \sin^2{\theta} \sin{\phi}$$ b₂ =

c₂ = c₁

$$r_{\phi} r_{\theta} \sin^2{\theta} + r r_{\theta \phi} \sin^2{\theta} + 2 r r_{\phi} \sin{\theta}\cos{\theta}$$ a₃ =

$$r r_{\theta}\sin^2{\theta} + r^2 \sin{\theta}\cos{\theta}$$ b₃ =

c₃ = c₁ (37)

The derivatives of the components of $\hat{n}$ with respect to $\theta$ are given by a formula similar to that for $\phi$:

$$\frac{\partial n_i}{\partial \theta} = S^{-1} \left [ a_i - b... ...ft ( \frac{r_{\theta}}{r} + \frac{c_i}{S^2} \right ) \right ]$$ (38)

where the functions a_i, b_i, and c_i are:

$$r_{\theta} r_{\phi} \sin{\phi} + r r_{\theta \phi} \sin{\phi} -r^... ...\cos{\theta}\cos{\phi} - r r_{\theta \theta} \sin{\theta}\cos{\theta}\cos{\phi}$$ a₁ =

$$- r r_{\theta} \cos^2{\theta}\cos{\phi} + r r_{\theta} \sin^2{\th... ... r_{\theta} \sin^2{\theta}\cos{\phi} + 2 r^2 \sin{\theta}\cos{\theta}\cos{\phi}$$

$$r r_{\phi} \sin{\phi} - r r_{\theta} \sin{\theta} \cos{\theta} \cos{\phi} + r^2 \sin^2{\theta} \cos{\phi}$$ b₁ =

$$r_{\phi} r_{\theta \phi} + r_{\theta} r_{\theta \theta} \sin^2{\t... ...{\theta} + r^2_{\theta} \sin{\theta}\cos{\theta} + r^2 \sin{\theta}\cos{\theta}$$ c₁ =

$$-r_{\phi} r_{\theta} \cos{\phi} - r r_{\theta \phi} \cos{\phi} -r... ...}\cos{\theta}\sin{\phi} -r r_{\theta \theta} \sin{\theta}\cos{\theta}\sin{\phi}$$ a₂ =

$$- r r_{\theta} \cos^2{\theta}\sin{\phi} + r r_{\theta} \sin^2{\th... ... r_{\theta} \sin^2{\theta}\sin{\phi} + 2 r^2 \sin{\theta}\cos{\theta}\sin{\phi}$$

$$- r r_{\phi} \cos{\phi} - r r_{\theta} \sin{\theta} \cos{\theta} \sin{\phi} + r^2 \sin^2{\theta} \sin{\phi}$$ b₂ =

c₂ = c₁

$$r^2_{\theta} \sin^2{\theta} + r r_{\theta \theta} \sin^2{\theta} +2 r r_{\theta} \sin{\theta}\cos{\theta}$$ a₃ =

$$+ 2 r r_{\theta} \sin{\theta}\cos{\theta} + r^2 \cos^2{\theta} - r^2 \sin^2{\theta}$$

$$r r_{\theta}\sin^2{\theta} + r^2 \sin{\theta}\cos{\theta}$$ b₃ =

c₃ = c₁ (39)