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@@ -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} |