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-rw-r--r--ElMag.tex259
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diff --git a/ElMag.tex b/ElMag.tex
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--- a/ElMag.tex
+++ b/ElMag.tex
@@ -1,6 +1,7 @@
% !TeX program = xelatex
% !TeX encoding = utf8
% !TeX root = ElMag.tex
+% vim: sw=2 ts=2 et:
%% TODO: publish to CTAN
\documentclass[margin=normal]{tex/hsrzf}
@@ -23,6 +24,9 @@
lang={english},
]{doclicense}
+%% Theorems
+\usepackage{amsthm}
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Metadata
@@ -37,6 +41,28 @@
\date{\thesemester}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Macros and settings
+
+%% Theorems
+\newtheoremstyle{elmagzf} % name of the style to be used
+ {\topsep}
+ {\topsep}
+ {}
+ {0pt}
+ {\bfseries}
+ {.}
+ { }
+ { }
+
+\theoremstyle{elmagzf}
+\newtheorem{theorem}{Theorem}
+\newtheorem{method}{Method}
+\newtheorem{application}{Application}
+\newtheorem{definition}{Definition}
+\newtheorem{remark}{Remark}
+\newtheorem{note}{Note}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Document
\begin{document}
@@ -56,7 +82,240 @@
% actual content
\clearpage
+\twocolumn
\setcounter{page}{1}
\pagenumbering{arabic}
+\section{Vector Analysis Recap}
+
+\begin{definition}[Gradient vector]
+ The \emph{gradient} of a function \(f(\vec{x}), \vec{x}\in\mathbb{R}^m\) is a
+ column vector containing the partial derivatives
+ in each direction.
+ \[
+ \grad f (\vec{x}) = \sum_{i=1}^m \partial_{x_i} f(\vec{x}) \vec{e}_i
+ = \begin{pmatrix}
+ \partial_{x_1} f(\vec{x}) \\
+ \vdots \\
+ \partial_{x_m} f(\vec{x}) \\
+ \end{pmatrix}
+ \]
+\end{definition}
+
+\begin{theorem}[Gradient in curvilinear coordinates]
+ Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a scalar field. In cylindrical coordinates \((r,\phi,z)\)
+ \[
+ \grad f = \uvec{r}\,\partial_r f
+ + \uvec{\phi}\,\frac{1}{r}\partial_\phi f
+ + \uvec{z}\,\partial_z f,
+ \]
+ and in spherical coordinates \((r,\theta,\phi)\)
+ \[
+ \grad f = \uvec{r}\,\partial_r f
+ + \uvec{\theta}\,\frac{1}{r} \partial_\theta f
+ + \uvec{\phi}\,\frac{1}{r \sin\theta} \partial_\phi f.
+ \]
+\end{theorem}
+
+\begin{definition}[Divergence]
+ Let \(\vec{F}: \mathbb{R}^m \to \mathbb{R}^m\) be a vector field.
+ The divergence of \(\vec{F} = (F_{x_1},\ldots, F_{x_m})^t\) is
+ \[
+ \div\vec{F} = \sum_{i = 1}^m \partial_{x_i} F_{x_i} ,
+ \]
+ as suggested by the (ab)use of the dot product notation.
+\end{definition}
+
+\begin{theorem}[Divergence in curvilinear coordinates]
+ Let \(\vec{F}: \mathbb{R}^3 \to \mathbb{R}^3\) be a field. In cylindrical coordinates \((r,\phi,z)\)
+ \[
+ \div \vec{F} = \frac{1}{r} \partial_r (r F_r)
+ + \frac{1}{r}\partial_\phi F_\phi
+ + \partial_z F_z,
+ \]
+ and in spherical coordinates \((r,\theta,\phi)\)
+ \begin{align*}
+ \div \vec{F} = \frac{1}{r^2} \partial_r (r^2 F_r)
+ & + \frac{1}{r \sin\theta} \partial_\theta (\sin\theta F_\theta) \\
+ & + \frac{1}{r \sin\theta} \partial_\phi F_\phi
+ \end{align*}
+\end{theorem}
+
+\begin{theorem}[Divergence theorem, Gauss's theorem]
+ Because the flux on the boundary \(\partial V\) of a volume \(V\) contains
+ information of the field inside of \(V\), it is possible relate the two with
+ \[
+ \int_V \div \vec{F} \,dv = \oint_{\partial V} \vec{F} \dotp d\vec{s} .
+ \]
+\end{theorem}
+
+\begin{definition}[Curl]
+ Let \(\vec{F}\) be a vector field. In 2 dimensions
+ \[
+ \curl \vec{F} = \left(\partial_x F_y - \partial_y F_x\right)\uvec{z}.
+ \]
+ And in 3D
+ \[
+ \curl \vec{F} = \begin{pmatrix}
+ \partial_y F_z - \partial_z F_y \\
+ \partial_z F_x - \partial_x F_z \\
+ \partial_x F_y - \partial_y F_x
+ \end{pmatrix}
+ = \begin{vmatrix}
+ \uvec{x} & \uvec{y} & \uvec{z} \\
+ \partial_x & \partial_y & \partial_z \\
+ F_x & F_y & F_z
+ \end{vmatrix} .
+ \]
+\end{definition}
+
+\begin{definition}[Curl in curvilinear coordinates]
+ Let \(\vec{F}: \mathbb{R}^3 \to \mathbb{R}^3\) be a field. In cylindrical coordinates \((r,\phi,z)\)
+ \begin{align*}
+ \curl \vec{F} =
+ &\left(\frac{1}{r} \partial_\phi F_z - \partial_z F_\phi \right) \uvec{r} \\
+ &+ \left(\partial_z F_r - \partial_r F_z \right) \uvec{\phi} \\
+ &+ \frac{1}{r} \bigg[
+ \partial_r (rF_\phi) - \partial_\phi F_r
+ \bigg] \uvec{z},
+ \end{align*}
+ and in spherical coordinates \((r,\theta,\phi)\)
+ \begin{align*}
+ \curl \vec{F} =
+ &\frac{1}{r \sin\theta} \bigg[
+ \partial_\theta (\sin\theta F_\phi) - \partial_\phi F_\theta
+ \bigg] \uvec{r} \\
+ &+ \frac{1}{r} \bigg[
+ \frac{1}{\sin\theta} \partial_\phi F_r - \partial_r (r F_\phi)
+ \bigg] \uvec{\theta} \\
+ &+ \frac{1}{r} \bigg[
+ \partial_r (r F_\theta) - \partial_\theta F_r
+ \bigg] \uvec{\phi} .
+ \end{align*}
+\end{definition}
+
+\begin{theorem}[Stokes' theorem]
+ \[
+ \int_\mathcal{S} \curl \vec{F} \dotp d\vec{s}
+ = \oint_{\partial\mathcal{S}} \vec{F} \dotp d\vec{r}
+ \]
+\end{theorem}
+
+\begin{definition}[Laplacian operator]
+ A second vector derivative is so important that it has a special name. For a
+ scalar function \(f: \mathbb{R}^m \to \mathbb{R}\) the divergence of the
+ gradient
+ \[
+ \laplacian f = \div (\grad f) = \sum_{i=1}^m \partial_{x_i}^2 f_{x_i}
+ \]
+ is called the \emph{Laplacian operator}.
+\end{definition}
+
+\begin{theorem}[Laplacian in curvilinear coordinates]
+ Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a scalar field. In cylindrical coordinates \((r,\phi,z)\)
+ \[
+ \laplacian f = \frac{1}{r} \partial_r (r \partial_r f)
+ + \frac{1}{r^2} \partial_\phi^2 f
+ + \partial_z^2 f
+ \]
+ and in spherical coordinates \((r,\theta,\phi)\)
+ \begin{align*}
+ \laplacian f =
+ \frac{1}{r^2} \partial_r ( r^2 \partial_r f)
+ & + \frac{1}{r^2\sin\theta} \partial_\theta (\sin\theta \partial_\theta f) \\
+ & + \frac{1}{r^2 \sin^2 \theta} \partial_\phi^2 f.
+ \end{align*}
+\end{theorem}
+
+\begin{definition}[Vector Laplacian]
+ The Laplacian operator can be extended on a vector field \(\vec{F}\) to the
+ \emph{Laplacian vector} by applying the Laplacian to each component:
+ \[
+ \vlaplacian \vec{F}
+ = (\laplacian F_x)\uvec{x}
+ + (\laplacian F_y)\uvec{y}
+ + (\laplacian F_z)\uvec{z} .
+ \]
+ The vector Laplacian can also be defined as
+ \[
+ \vlaplacian \vec{F} = \grad (\div \vec{F}) - \curl (\curl \vec{F}).
+ \]
+\end{definition}
+
+\begin{theorem}[Product rules and second derivatives]
+ Let \(f,g\) be sufficiently differentiable scalar functions \(D
+ \subseteq\mathbb{R}^m \to \mathbb{R}\) and \(\vec{A}, \vec{B}\) be
+ sufficiently differentiable vector fields in \(\mathbb{R}^m\) (with \(m = 2\)
+ or 3 for equations with the curl).
+ \begin{itemize}
+ \item Rules with the gradient
+ \begin{align*}
+ \grad (\div \vec{A}) &= \curl \curl \vec{A} + \vlaplacian \vec{A} \\
+ \grad (f\cdot g) &= (\grad f)\cdot g + f\cdot \grad g \\
+ \grad (\vec{A} \dotp \vec{B}) &=
+ (\vec{A} \dotp \grad) \vec{B}
+ + (\vec{B} \dotp \grad) \vec{A} \\
+ & + \vec{A} \crossp (\curl \vec{B})
+ + \vec{B} \crossp (\curl \vec{A})
+ \end{align*}
+ \item Rules with the divergence
+ \begin{align*}
+ \div (\grad f) &= \laplacian f \\
+ \div (\curl \vec{A}) &= 0 \\
+ \div (f\cdot \vec{A}) &= (\grad f) \dotp \vec{A} + f\cdot (\div \vec{A}) \\
+ \div (\vec{A}\crossp\vec{B}) &= (\curl \vec{A})\dotp \vec{B}
+ - \vec{A} \cdot (\curl\vec{B})
+ \end{align*}
+ \item Rules with the curl
+ \begin{align*}
+ \curl (\grad f) &= \vec{0} \\
+ \curl (\curl \vec{A}) &= \grad (\div \vec{A}) - \vlaplacian \vec{A} \\
+ \curl (\vlaplacian \vec{A}) &= \vlaplacian (\curl \vec{A}) \\
+ \curl (f\cdot \vec{A}) &= (\grad f)\crossp \vec{A} + f\cdot \curl \vec{A} \\
+ \curl (\vec{A}\crossp\vec{B}) &=
+ (\vec{B} \dotp \grad) \vec{A} - (\vec{A} \dotp \grad) \vec{B} \\
+ &+ \vec{A} \dotp (\div \vec{B}) - \vec{B} \dotp (\div \vec{A})
+ \end{align*}
+ \end{itemize}
+\end{theorem}
+
+\section{Electrodynamics Recap}
+
+\subsection{Maxwell's equations}
+
+Maxwell's equations in matter in their integral form are
+\begin{subequations}
+ \begin{align}
+ \oint_{\partial S} \vec{E} \dotp d\vec{l} &= -\frac{d}{dt} \int_S \vec{B} \dotp d\vec{s}, \\
+ \oint_{\partial S} \vec{H} \dotp d\vec{l} &= \int_S \left(
+ \vec{J} + \partial_t \vec{D}
+ \right) \dotp d\vec{s}, \\
+ \oint_{\partial V} \vec{D} \dotp d\vec{s} &= \int_V \rho \,dv, \\
+ \oint_{\partial V} \vec{B} \dotp d\vec{s} &= 0.
+ \end{align}
+\end{subequations}
+Where \(\vec{J}\) and \(\rho\) are the \emph{free current density} and \emph{free charge density} respectively.
+
+\subsection{Isotropic linear materials and boundary conditions}
+
+Inside of so called isotropic linear materials the fields and flux (or current) densities are proportional, i.e.
+\begin{align*}
+ \vec{D} &= \varepsilon \vec{E}, & \vec{J} &= \sigma \vec{E}, & \vec{B} &= \mu \vec{H}.
+\end{align*}
+
+Between two materials (1) and (2) the following boundary conditions must be satisfied:
+\begin{align*}
+ &\uvec{n} \dotp \vec{D}_1 = \uvec{n} \dotp \vec{D}_2 + \rho_s &
+ &\uvec{n} \crossp \vec{E}_1 = \uvec{n} \crossp \vec{E}_2 \\
+ &\uvec{n} \dotp \vec{J}_1 = \uvec{n} \dotp \vec{J}_2 - \partial_t \rho_s &
+ &\uvec{n} \crossp \vec{H}_1 = \uvec{n} \crossp \vec{H}_2 + \vec{J}_s \\
+ &\uvec{n} \dotp \vec{B}_1 = \uvec{n} \dotp \vec{B}_2 - \partial_t \rho_s &
+ &\uvec{n} \crossp \vec{M}_1 = \uvec{n} \crossp \vec{M}_2 + \vec{J}_{s,m}
+\end{align*}
+
+\subsection{Magnetic vector potential}
+
+\section{Laplace and Poisson's equation}
+
+
\end{document}