Appendix 1: Physical constants Table a1.1 Physical constants [7 ] Appendix 2: Units and Their Conversions Table a2.2 Units and Their Conversions Appendix 3: Selections From Vector Analysis Vector Algebra
$$\begin{aligned} \vec {a}{\cdot } \vec {b}=\sum _\alpha ^{x,y,z} {a_\alpha b_\alpha }. \end{aligned}$$
(a3.1)
$$\begin{aligned}&\vec {a}\times \vec {b}=\left| {{\begin{array}{lll} {\vec {e}_x }&{\vec {e}_y }&{\vec {e}_z } \\ {a_x }&{a_y }&{a_z } \\ {b_x }&{b_y }&{b_z } \\ \end{array} }} \right|=\left(a_y b_z -a_z b_y \right)\vec {e}_x +\left(a_z b_x -a_x b_z \right)\vec {e}_y +\left(a_x b_y -a_y b_x \right)\vec {e}_z.\nonumber \\&\left(\vec {e}_\alpha : \text{ unit} \text{ vector} \text{ in} \text{ the}\;{\alpha }\;\mathrm{ direction}\right) \end{aligned}$$
(a3.2)
$$\begin{aligned} \vec {a}{\cdot } \bigl (\vec {b}\times \vec {c} \bigr )=\vec {b}{\cdot } \bigl (\vec {c}\times \vec {a}\bigr )=\vec {c}{\cdot } \bigl (\vec {a}\times \vec {b}\bigr )=\left| {{\begin{array}{lll} {a_x }&{a_y }&{a_z } \\ {b_x }&{b_y }&{b_z } \\ {c_x }&{c_y }&{c_z } \\ \end{array} }} \right|. \end{aligned}$$
(a3.3)
$$\begin{aligned} \vec {a}\times \bigl (\vec {b}\times \vec {c}\bigr )=\vec {b} \bigl (\vec {a}{\cdot }\vec {c} \bigr )-\vec {c}\bigl (\vec {a}{\cdot } \vec {b}\bigr ). \end{aligned}$$
(a3.4)
Del, Gradient, Divergence, and Curl Operators
$$\begin{aligned} \nabla \equiv \sum _\alpha ^{x,y,z} \vec {e}_\alpha \frac{\partial }{\partial \alpha }. \qquad (\text{ Del} \text{ or} \text{ Nabla} \text{ operator}) \end{aligned}$$
(a3.5)
$$\begin{aligned} \nabla {\cdot } \nabla =\nabla ^{2}=\sum _\alpha ^{x,y,z} {\frac{\partial ^{2}}{\partial \alpha ^{2}}} . \qquad (\text{ Laplace} \text{ operator}) \end{aligned}$$
(a3.6)
$$\begin{aligned} \nabla \varphi \equiv (\frac{d\varphi }{d\vec {l}})_{\max } =\sum _\alpha ^{x,y,z} {\frac{\partial \varphi }{\partial \alpha }\vec {e}_\alpha } . \qquad (\text{ gradient} \text{ of}\;{\varphi }) \end{aligned}$$
(a3.7)
$$\begin{aligned} \nabla {\cdot } \vec {a}\equiv \lim _{v\rightarrow 0} \frac{\oint {\vec {a}{\cdot } d\vec {S}} }{v}=\sum _\alpha ^{x,y,z}{\frac{\partial a_\alpha }{\partial \alpha }} . \qquad (\text{ divergence} \text{ of}\;{\vec {a}}, v: \\ \text{ volume} \text{ surrounded} \text{ by} \text{ the} \text{ closed} \text{ surface} \text{ S}) \end{aligned}$$
(a3.8)
$$\begin{aligned} \nabla \times \vec {a}\equiv \lim _{S\rightarrow 0} \left(\frac{\oint {\vec {a}{\cdot } \mathrm{ d}\vec {l}} }{\vec {S}}\right)_{\max } =\left| {{\begin{array}{lll} {\vec {e}_x }&{\vec {e}_y }&{\vec {e}_z } \\ {\frac{\partial }{\partial x}}&{\frac{\partial }{\partial y}}&{\frac{\partial }{\partial z}} \\ {a_x }&{a_y }&{a_z } \\ \end{array} }} \right|. \qquad (\text{ curl} \text{ of}\;\vec {a}, S{:}\ \text{ area} \text{ of} \text{ the} \text{ closed} \text{ curve}\;l) \end{aligned}$$
(a3.9)
$$\begin{aligned} \nabla \times \nabla \varphi =0. \end{aligned}$$
(a3.10)
$$\begin{aligned} \nabla {\cdot } \nabla \times \vec {a}=0. \end{aligned}$$
(a3.11)
$$\begin{aligned} \nabla \times \nabla \times \vec {a}=\left(\nabla \nabla {\cdot } -\nabla ^{2}\right)\vec {a}. \end{aligned}$$
(a3.12)
If \(\nabla \times \vec {a}=0\) in region D, \(\vec {a}\) is a divergence of a scalar:
$$\begin{aligned} \vec {a}=\nabla \varphi . \end{aligned}$$
(a3.13)
If \(\nabla {\cdot } \vec {a}=0\) in region D, \(\vec {a}\) is a curl of a vector:
$$\begin{aligned} \vec {a}=\nabla \times \vec {A}. \end{aligned}$$
(a3.14)
Del Operations on Products of Two Functions
$$\begin{aligned} \nabla \bigl (\vec {a}{\cdot } \vec {b}\bigr )=\vec {a}\times \bigl (\nabla \times \vec {b} \bigr )+\vec {b}\times \bigl (\nabla \times \vec {a} \bigr )+\bigl (\vec {a}{\cdot } \nabla \bigr )\vec {b}+\bigl (\vec {b}{\cdot } \nabla \bigr )\vec {a}. \end{aligned}$$
(a3.15)
$$\begin{aligned} \nabla {\cdot } \bigl (\varphi \vec {a}\bigr )=\bigl (\nabla \varphi \bigr ){\cdot } \vec {a}+\varphi \nabla {\cdot } \vec {a}. \end{aligned}$$
(a3.16)
$$\begin{aligned} \nabla {\cdot } \bigl (\vec {a}\times \vec {b}\bigr )=\bigl (\nabla \times \vec {a}\bigr ){\cdot } \vec {b}-\vec {a}{\cdot } \bigl (\nabla \times \vec {b}\bigr ). \end{aligned}$$
(a3.17)
$$\begin{aligned} \nabla \times \bigl (\varphi \vec {a}\bigr )=\bigl (\nabla \varphi \bigr )\times \vec {a}+\varphi \nabla \times \vec {a}. \end{aligned}$$
(a3.18)
$$\begin{aligned} \nabla \times \bigl (\vec {a}\times \vec {b}\bigr )=\bigl (\vec {b}{\cdot } \nabla \bigr )\vec {a}-\bigl (\vec {a}{\cdot } \nabla \bigr )\vec {b}+\bigl (\nabla {\cdot } \vec {b}\bigr )\vec {a}-\bigl (\nabla {\cdot } \vec {a}\bigr )\vec {b}. \end{aligned}$$
(a3.19)
Del Operations in Spherical Coordinate System
$$\begin{aligned} \nabla \varphi =\frac{\partial \varphi }{\partial r}\vec {e}_r +\frac{1}{r}\frac{\partial \varphi }{\partial \theta }\vec {e}_\theta +\frac{1}{r\sin \theta }\frac{\partial \varphi }{\partial \phi }\vec {e}_\phi . \end{aligned}$$
(a3.20)
$$\begin{aligned} \nabla {\cdot } \vec {a}=\frac{1}{r^{2}}\frac{\partial \bigl (a_r r^{2}\bigr )}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial (a_\theta \sin \theta )}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial a_\phi }{\partial \phi }. \end{aligned}$$
(a3.21)
$$\begin{aligned} \nabla \times \vec {a}&=\frac{1}{r\sin \theta }\left[\frac{\partial \bigl (a_\phi \sin \theta \bigr )}{\partial \theta }-\frac{\partial a_\theta }{\partial \phi }\right]\vec {e}_r +\frac{1}{r}\left[\frac{1}{\sin \theta }\frac{\partial a_r }{\partial \phi }-\frac{\partial (a_\phi r)}{\partial r}\right]\vec {e}_\theta \nonumber \\&\quad +\frac{1}{r}\left[\frac{\partial (a_\theta r)}{\partial r}-\frac{\partial a_r }{\partial \theta }\right]\vec {e}_\phi . \end{aligned}$$
(a3.22)
$$\begin{aligned} \nabla ^{2}\varphi =\frac{1}{r^{2}}\frac{\partial }{\partial r}\bigl (r^{2}\frac{\partial \varphi }{\partial r}\bigr )+\frac{1}{r^{2}\sin \theta }\frac{\partial }{\partial \theta } \bigl (\sin \theta \frac{\partial \varphi }{\partial \theta } \bigr )+\frac{1}{\left(r\sin \theta \right)^{2}}\frac{\partial ^{2}\varphi }{\partial \phi ^{2}}. \end{aligned}$$
(a3.23)
Integral Relations
$$\begin{aligned} \int {\mathrm{ d}\vec {S}\times \nabla \varphi } =\oint {\varphi \mathrm{ d}\vec {l}} . \end{aligned}$$
(a3.24)
$$\begin{aligned} \int {\nabla {\cdot } \vec {a}\mathrm{ d}v} =\oint {\vec {a}{\cdot } \mathrm{ d}\vec {S}} . \end{aligned}$$
(a3.25)
$$\begin{aligned} \int {\nabla \times \vec {a}\mathrm{ d}v=\oint {\mathrm{ d}\vec {S}\times \vec {a}} } . \end{aligned}$$
(a3.26)
$$\begin{aligned} \int {\bigl (\nabla \times \vec {a}\bigr ){\cdot } \mathrm{ d}\vec {S}=\oint {\vec {a}{\cdot } \mathrm{ d}\vec {l}} } . \end{aligned}$$
(a3.27)
$$\begin{aligned} \int {\left[\psi \nabla ^{2}\varphi +(\nabla \psi ){\cdot } (\nabla \varphi )\right]\mathrm{ d}v=\oint {\psi (\nabla \varphi ){\cdot } \mathrm{ d}\vec {S}} }. \end{aligned}$$
(a3.28)
$$\begin{aligned} \int {\left(\psi \nabla ^{2}\varphi -\varphi \nabla ^{2}\psi \right)} \mathrm{ d}v=\oint {\left(\psi \nabla \varphi -\varphi \nabla \psi \right){\cdot } \mathrm{ d}\vec {S}} . \end{aligned}$$
(a3.29)
