$\renewcommand{\Re}{\operatorname{Re}}$
$\renewcommand{\Im}{\operatorname{Im}}$
$\newcommand{\erf}{\operatorname{erf}}$
$\newcommand{\dag}{\dagger}$
$\newcommand{\const}{\mathrm{const}}$
#[Chapter A. Appendices](id:chapter-A)
##[A.1. Field theory](id:sect-A.1)
__________
> 1. [Green, Gauss, Stokes formulae](#sect-A.1.1)
> 2. [Properties of $\nabla$](#sect-A.1.2)
###[Green, Gauss, Stokes formulae](id:sect-A.1.1)
Let $D$ be a bounded domain in $\mathbb{R}^2$ and $L=\partial D$ be its boundary. Then
\begin{equation}
-\int\_{L} \mathbf{A}\cdot \mathbf{n} \,ds= \iint_D (\nabla \cdot \mathbf{A})\,dS
\label{eq-A.1.1}
\end{equation}
where the left-hand side expression is a linear integral, the right-hand side expression is an area integral and $\mathbf{n}$ is a unit inner normal to $L$. This is *Green formula*.
Let $V$ be a bounded domain in $\mathbb{R}^3$ and $\Sigma=\partial V$ be its boundary. Then
\begin{equation}
-\iint\_{\Sigma} \mathbf{A}\cdot \mathbf{n} \,dS= \iiint_D (\nabla \cdot \mathbf{A})\,dV
\label{eq-A.1.2}
\end{equation}
where the left-hand side expression is a surface integral, the right-hand side expression is a volume integral and $\mathbf{n}$ is a unit inner normal to $\Sigma$. This is *Gauss formula*.
**[Remark 1.](id:remark-A.1.1)**
a. Here sign "$-$"appears because $\mathbf{n}$ is a unit *inner* normal.
b. Gauss formula holds in any dimension. It also holds in any "straight" coordinate system.
c. In the curvilinear coordinate system $(u\_1,\ldots, u\_n)$ in also holds but divergence must be calculated as
\begin{equation}
\nabla \cdot \mathbf{A}= \sum\_k J^{-1}\partial\_{u\_k} (A^k J)=
\sum\_k \partial\_{u\_k}A^k + \sum\_k (\partial\_{u\_k} \ln J) \, A^k
\label{eq-A.1.3}
\end{equation}
where $Jdu\_1\ldots du\_n$ is a volume form in these coordinates. F.e. in the spherical coordinates if $\mathbf{A}$ is radial, $\mathbf{A}= A\_r \mathbf{r}/r$ (with $r=|\mathbf{r}|$ we have
$\partial\_{r} \ln J)= (n-1)r^{-1}$ and therefore $\nabla\cdot A\_r \mathbf{r}/r= \partial\_r A\_r + (n-1)r^{-1}A\_r$.
Let $D$ be a bounded domain in $\mathbb{R}^2$ and $L=\partial D$ be its boundary, counter–clockwise oriented (if $L$ has several components then inner components should be clockwise oriented). Then
\begin{equation}
\oint\_{L} \mathbf{A}\cdot \, d\, \mathbf{r}= \iint_D (\nabla \times \mathbf{A})\cdot\mathbf{n}\,dS
\label{eq-A.1.4}
\end{equation}
where the left-hand side expression is a line integral, the right-hand side expression is an area integral and $\mathbf{n}=\mathbf{k}$. This is *Green formula* again.
Let $\Sigma$ be a bounded piece of the surface in $\mathbb{R}^3$ and $L=\partial \Sigma$ be its boundary. Then
\begin{equation}
\oint\_{L} \mathbf{A}\cdot \, d\, \mathbf{l}= \iint\_\Sigma (\nabla \times \mathbf{A})\cdot\mathbf{n}\,dS
\label{eq-A.1.5}
\end{equation}
where the left-hand side expression is a line integral, the right-hand side expression is a surface integral and $\mathbf{n}$ is a unit normal to $\Sigma$; orientation of $L$ should match to direction of $\mathbf{n}$.
This is *Stokes formula*.
**[Remark 2.](id:remark-A.1.2)**
a. We can describe orientation in the Green formula as "the pair $\\{d\mathbf{r}, \mathbf{n}\\}$ has a right-hand orientation"
b. We can describe orientation in the Stokes formula as "the triple $\\{d\mathbf{r}, \boldsymbol{\nu}, \mathbf{n}\\}$ has a right-hand orientation" where $\boldsymbol{\nu}$ is a normal to $L$ which is tangent to $\Sigma$ and directed inside of $\Sigma$.
c. Stokes formula holds in any dimension of the surface $\Sigma$ but then it should be formulated in terms of differential forms \begin{equation} \int \_\Sigma d\omega = \int \_{\partial\Sigma}\omega\tag{Stokes formula}\end{equation} which is the material of Analysis II class (aka Calculus II Pro).
###[Properties of $\nabla$](id:sect-A.1.2)
####[Definitions](id:sect-A.1.2.1)
**[Definition 1.](id:def-A.1.1)**
a. Operator $\nabla$ is defined as
\begin{equation}
\nabla = \mathbf{i} \partial\_x + \mathbf{j} \partial\_y+ \mathbf{k} \partial\_z.
\label{eq-A.1.6}
\end{equation}
b. It could be applied to a scalar function resulting in its gradient ($\operatorname{grad}\phi$)
\begin{equation\*}
\nabla \phi = \mathbf{i} \partial\_x\phi + \mathbf{j} \partial\_y\phi+ \mathbf{k} \partial\_z\phi
\end{equation\*}
c. and to vector function $\mathbf{A}=A\_x\mathbf{i}+A\_y\mathbf{j}+A\_z\mathbf{k}$ resulting in its divergence ($\operatorname{div}\mathbf{A}$)
\begin{equation\*}
\nabla \cdot \mathbf{A} = \partial\_xA\_x + \partial\_y A\_y+ \partial\_zA\_z
\end{equation\*}
d. and also in its curl ($\operatorname{curl}\mathbf{A}$) or rotor ($\operatorname{rot}\mathbf{A}$), depending on the mathematical tradition:
\begin{equation\*}
\nabla \times \mathbf{A} =
\left| \begin{matrix} \mathbf{i} & \mathbf{j} &\mathbf{k} \\\\
\partial\_x & \partial\_y & \partial\_z\\\\
A\_x & A\_y &A\_z\end{matrix}\right|
\end{equation\*}
which is equal to
\begin{equation\*}
(\partial\_y A\_z-\partial\_z A\_y)\mathbf{i}+
(\partial\_z A\_x-\partial\_x A\_z)\mathbf{j}+
(\partial\_x A\_y-\partial\_y A\_x)\mathbf{k}.
\end{equation\*}
####[Double application](id:sect-A.1.2.2)
**[Definition 2.](id:def-A.1.2)**
\begin{equation}
\Delta= \nabla^2 = \nabla\cdot \nabla=
\partial\_x^2 + \partial\_y^2+ \partial\_z^2.
\label{eq-A.1.7}
\end{equation}
is *Laplace operator* or simply *Laplacian*.
Four formulae to remember:
\begin{gather}
\nabla (\nabla \phi)= \Delta \phi,\label{eq-A.1.8}\\\\[3pt]
\nabla \times (\nabla \phi)= 0,\label{eq-A.1.9}\\\\[3pt]
\nabla \cdot (\nabla \times \mathbf{A})= 0,\label{eq-A.1.10}\\\\[3pt]
\nabla \times (\nabla \times \mathbf{A})=
-\Delta \mathbf{A} + \nabla (\nabla \cdot \mathbf{A}) \label{eq-A.1.11}
\end{gather}
where all but the last one are obvious and the last one follows from
\begin{equation}
\mathbf{a}\times (\mathbf{a} \times \mathbf{b})= - \mathbf{a}^2 \mathbf{b}+ (\mathbf{a}\cdot \mathbf{b}) \mathbf{a}
\label{eq-A.1.12}
\end{equation}
which is the special case of
\begin{equation}
\mathbf{a}\times (\mathbf{b} \times \mathbf{c})= \mathbf{b}(\mathbf{a}\cdot\mathbf{c})-
\mathbf{c}(\mathbf{a}\cdot\mathbf{b}).
\label{eq-A.1.13}
\end{equation}
####[Application to the product](id:sect-A.1.2.3)
Recall *Leibniz rule* how to apply the first derivative to the product which can be symbolically written as
\begin{equation\*}
\partial (uv)=
(\partial\_u + \partial\_v)(uv)= \partial\_u (uv)+\partial\_v (uv)=
v\partial\_u (u) +u\partial\_v (v)=v\partial u +u\partial v
\end{equation\*}
where subscripts "$u$" or "$v$" mean that it should be applied to $u$ or $v$ only.
Since $\nabla$ is a linear combination of the first derivatives, it inherits the same rule. Three formulae are easy
\begin{gather}
\nabla ( \phi\psi)= \phi\nabla \psi +\psi \nabla \phi,\label{eq-A.1.14}\\\\[3pt]
\nabla \cdot ( \phi\mathbf{A})=
\phi\nabla \cdot \mathbf{A} +\nabla \phi\cdot \mathbf{A} , \label{eq-A.1.15}\\\\[3pt]
\nabla \times ( \phi\mathbf{A})=
\phi\nabla \times \mathbf{A} +\nabla \phi\times \mathbf{A} , \label{eq-A.1.16}\end{gather}
and the fourth follows from the Leibniz rule and (\ref{eq-A.1.13})
\begin{equation}
\nabla \times ( \mathbf{A}\times \mathbf{B})=
(\mathbf{B}\cdot\nabla)A-\mathbf{B}(\nabla\cdot \mathbf{A}) -
(\mathbf{A}\cdot\nabla)B+\mathbf{A}(\nabla\cdot \mathbf{B}). \label{eq-A.1.17}
\end{equation}
_________
[$\Leftarrow$](../Chapter14/S14.2.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](./SA.2.html)