$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ #Chapter A. Appendices ##A.1. Field theory __________ > 1. [Green, Gauss, Stokes formulae](#sect-A.1.1) > 2. [Properties of $\nabla$](#sect-A.1.2) ###Green, Gauss, Stokes formulae 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.** 1. Here sign "$-$"appears because $\mathbf{n}$ is a unit *inner* normal. 2. Gauss formula holds in any dimension. It also holds in any "straight" coordinate system. 3. 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.** 1. We can describe orientation in the Green formula as "the pair $\\{d\mathbf{r}, \mathbf{n}\\}$ has a right-hand orientation" 2. 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 $\Sigma$. 3. 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 $\nabla$ ####Definitions **Definition 1.** 1. 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} 2. 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\*} 3. 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\*} 4. 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 **Definition 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 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/S14.4.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](./SA.2.html)