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authorNao Pross <np@0hm.ch>2022-05-19 18:58:44 +0200
committerNao Pross <np@0hm.ch>2022-05-19 18:58:44 +0200
commitcfd03547b8f392701471bd25aa98926494c923c0 (patch)
tree087145951b3fb505116f63ad90670eb215c87dac
parentderivation of the orthogonal basis functions using the laplacian operator (diff)
downloadFourierOnS2-cfd03547b8f392701471bd25aa98926494c923c0.tar.gz
FourierOnS2-cfd03547b8f392701471bd25aa98926494c923c0.zip
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diff --git a/presentation/KugelBSc.bib b/presentation/KugelBSc.bib
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@@ -0,0 +1,119 @@
+
+@article{carvalhaes_surface_2015,
+ title = {The surface Laplacian technique in {EEG}: Theory and methods},
+ volume = {97},
+ issn = {01678760},
+ url = {https://linkinghub.elsevier.com/retrieve/pii/S0167876015001749},
+ doi = {10.1016/j.ijpsycho.2015.04.023},
+ shorttitle = {The surface Laplacian technique in {EEG}},
+ pages = {174--188},
+ number = {3},
+ journaltitle = {International Journal of Psychophysiology},
+ shortjournal = {International Journal of Psychophysiology},
+ author = {Carvalhaes, Claudio and de Barros, J. Acacio},
+ urldate = {2022-05-16},
+ date = {2015-09},
+ langid = {english},
+ file = {Submitted Version:/Users/npross/Zotero/storage/VYU5MGWE/Carvalhaes and de Barros - 2015 - The surface Laplacian technique in EEG Theory and.pdf:application/pdf},
+}
+
+@article{ries_role_2013,
+ title = {Role of the lateral prefrontal cortex in speech monitoring},
+ volume = {7},
+ issn = {1662-5161},
+ url = {http://journal.frontiersin.org/article/10.3389/fnhum.2013.00703/abstract},
+ doi = {10.3389/fnhum.2013.00703},
+ journaltitle = {Frontiers in Human Neuroscience},
+ shortjournal = {Front. Hum. Neurosci.},
+ author = {Riès, Stephanie K. and Xie, Kira and Haaland, Kathleen Y. and Dronkers, Nina F. and Knight, Robert T.},
+ urldate = {2022-05-16},
+ date = {2013},
+ file = {Full Text:/Users/npross/Zotero/storage/FQHC9R5W/Riès et al. - 2013 - Role of the lateral prefrontal cortex in speech mo.pdf:application/pdf},
+}
+
+@inproceedings{schmitz_using_2012,
+ location = {Santa Clara, {CA}, {USA}},
+ title = {Using spherical harmonics for modeling antenna patterns},
+ isbn = {978-1-4577-1155-8 978-1-4577-1153-4 978-1-4577-1154-1},
+ url = {http://ieeexplore.ieee.org/document/6175298/},
+ doi = {10.1109/RWS.2012.6175298},
+ eventtitle = {2012 {IEEE} Radio and Wireless Symposium ({RWS})},
+ pages = {155--158},
+ booktitle = {2012 {IEEE} Radio and Wireless Symposium},
+ publisher = {{IEEE}},
+ author = {Schmitz, Arne and Karolski, Thomas and Kobbelt, Leif},
+ urldate = {2022-05-16},
+ date = {2012-01},
+}
+
+@artwork{depiep_electron_2013,
+ title = {Electron shell 001 Hydrogen (diatomic nonmetal)},
+ url = {https://commons.wikimedia.org/wiki/File:Electron_shell_001_Hydrogen_(diatomic_nonmetal)_-_no_label.svg},
+ shorttitle = {English},
+ author = {{DePiep}},
+ urldate = {2022-05-18},
+ date = {2013-08-14},
+ file = {Wikimedia Snapshot:/Users/npross/Zotero/storage/F99YS2EX/FileElectron_shell_001_Hydrogen_(diatomic_nonmetal)_-_no_label.html:text/html},
+}
+
+@artwork{baburov__2009,
+ title = {Русский: Процесс регистрации электроэнцефалографии},
+ url = {https://commons.wikimedia.org/wiki/File:Eeg_registration.jpg},
+ shorttitle = {Русский},
+ author = {{Baburov}},
+ urldate = {2022-05-19},
+ date = {2009-08-21},
+ file = {Wikimedia Snapshot:/Users/npross/Zotero/storage/PPG8LMTG/FileEeg_registration.html:text/html},
+}
+
+@artwork{maschen_divergence_2013,
+ title = {Divergence theorem in {EM}},
+ url = {https://commons.wikimedia.org/wiki/File:Divergence_theorem_in_EM.svg},
+ shorttitle = {English},
+ author = {{Maschen}},
+ urldate = {2022-05-19},
+ date = {2013-05-12},
+ file = {Wikimedia Snapshot:/Users/npross/Zotero/storage/Q6UC6RS8/FileDivergence_theorem_in_EM.html:text/html},
+}
+
+@video{minutephysics_better_2021,
+ title = {A Better Way To Picture Atoms},
+ url = {https://www.youtube.com/watch?v=W2Xb2GFK2yc},
+ abstract = {Thanks to Google for sponsoring a portion of this video!
+Support {MinutePhysics} on Patreon: http://www.patreon.com/minutephysics
+
+This video is about using Bohmian trajectories to visualize the wavefunctions of hydrogen orbitals, rendered in 3D using custom python code in Blender.
+
+{REFERENCES}
+A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I
+David Bohm, Physical Review, Vol 85 No. 2, January 15, 1952
+
+Speakable and Unspeakable in Quantum Mechanics
+J. S. Bell
+
+Trajectory construction of Dirac evolution
+Peter Holland
+
+The de Broglie-Bohm Causal Interpretation of Quantum Mechanics and its Application to some Simple Systems by Caroline Colijn
+
+Bohmian Trajectories as the Foundation of Quantum Mechanics
+http://arxiv.org/abs/0912.2666v1
+
+The Pilot-Wave Perspective on Quantum Scattering and Tunneling
+http://arxiv.org/abs/1210.7265v2
+
+A Quantum Potential Description of One-Dimensional Time-Dependent Scattering From Square Barriers and Square Wells
+Dewdney, Foundations of Physics, {VoL} 12, No. 1, 1982
+
+Link to Patreon Supporters: http://www.minutephysics.com/supporters/
+
+{MinutePhysics} is on twitter - @minutephysics
+And facebook - http://facebook.com/minutephysics
+
+Minute Physics provides an energetic and entertaining view of old and new problems in physics -- all in a minute!
+
+Created by Henry Reich},
+ author = {{minutephysics}},
+ urldate = {2022-05-19},
+ date = {2021-05-19},
+} \ No newline at end of file
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+\documentclass[xetex, onlymath]{beamer}
+\usefonttheme{serif}
+\usetheme{hsr}
+
+% use lmodern for math
+\usepackage{lmodern}
+
+% math packages
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{bm}
+
+\renewcommand{\vec}[1]{\mathbf{\bm{#1}}}
+
+% use plex font for monospaced, roboto for the rest
+\usepackage[T1]{fontenc}
+\usepackage{plex-otf} % monospaced
+% \usepackage{roboto} % other
+\renewcommand*\familydefault{\sfdefault}
+
+\usepackage{graphicx}
+\usepackage{booktabs}
+\usepackage{array}
+
+% biblopgraphy
+\usepackage[backend=bibtex, style=ieee]{biblatex}
+\addbibresource{KugelBSc.bib}
+
+% links
+\usepackage{hyperref}
+\hypersetup{
+ % Remove ugly boxes
+ hidelinks,
+ % Set colors
+ colorlinks = true,
+ anchorcolor = black,
+ citecolor = black,
+ filecolor = black,
+ linkcolor = black,
+ menucolor = black,
+ runcolor = black,
+ urlcolor = {black!50!blue},
+}
+
+% pretty drawings
+\usepackage{tikz}
+\usetikzlibrary{calc}
+\usepackage{xcolor}
+
+% source code
+\usepackage{listings}
+%% create a lstlisting style
+\lstdefinestyle{samplestyle}{
+ belowcaptionskip=\baselineskip,
+ breaklines=true,
+ frame=none,
+ inputencoding=utf8,
+ % margin
+ xleftmargin=\parindent,
+ % background
+ backgroundcolor=\color{hsr-lightgrey20},
+ % default language:
+ language=[LaTeX]TeX,
+ showstringspaces=false,
+ % font
+ basicstyle=\ttfamily\small,
+ identifierstyle=\color{hsr-black},
+ keywordstyle=\color{hsr-blue},
+ commentstyle=\color{hsr-black40},
+ stringstyle=\color{hsr-mauve80},
+}
+
+%% and set the chosen style
+\lstset{style=samplestyle, escapechar=`}
+
+% metadata
+\title{Spherical Harmonics}
+\author[NaoPross]{\texttt{Naoki Pross, Manuel Cattaneo}}
+\date{Spring Semester 2022}
+
+\institute[OST]{OST FHO Campus Rapperswil}
+% \logo{\includegraphics[width=3cm]{figs/hsr-logo}}
+
+\AtBeginSection[]
+{
+ \begin{frame}
+ \frametitle{Table of Contents}
+ \tableofcontents[currentsection]
+ \end{frame}
+}
+
+
+\begin{document}
+
+\frame{
+ \maketitle
+}
+
+\begin{frame}{Goals for Today}
+ \Large \uncover<1->{\textbf{Spherical Harmonics}} \uncover<2->{\,\textit{and}\, \textbf{Electron Orbitals}}
+ \begin{tikzpicture}
+ \uncover<1->{
+ \node (i1) {
+ \includegraphics[height=8cm, trim=200 100 50 50, clip]{figures/buchcover}
+ };
+ }
+
+ \uncover<2->{
+ \node (i2) at ($(i1) + (2cm, 0)$) {
+ \nocite{minutephysics_better_2021}
+ \includegraphics[height=65mm]{figures/orbitals-minutephysics}
+ };
+ }
+ \end{tikzpicture}
+\end{frame}
+
+\section{Fourier on \(\mathbb{R}^2\)}
+
+\begin{frame}{Nice Periodic Functions}
+ \begin{definition}
+ A function
+ \[
+ f : \mathbb{R}^2 \to \mathbb{C}
+ \]
+ is a ``nice periodic function'' when it is
+ \begin{itemize}
+ \item smooth,
+ \item differentiable,
+ \item \textcolor{gray}{(abs.)} integrable,
+ \item periodic on \([0, 1] \times [0, 1]\), i.e.
+ \[
+ f(\mu, \nu) = f(\mu + 1, \nu) = f(\mu, \nu + 1).
+ \]
+ \end{itemize}
+ \end{definition}
+\end{frame}
+
+\begin{frame}{Function Space}
+ \begin{block}{Basis Functions}
+ The space of nice periodic functions is spanned by the (also nice) functions
+ \[
+ B_{m, n}(\mu, \nu) = e^{i2\pi m\mu} e^{i2\pi n\nu}.
+ \]
+ \end{block}
+\end{frame}
+
+\begin{frame} \centering
+ \includegraphics[height=.9\paperheight]{figures/flat-basis-functions}
+\end{frame}
+
+\begin{frame}{Inner Product}
+ \begin{definition}<1->
+ Let \(f(\mu, \nu)\) and \(g(\mu, \nu)\) be nice periodic functions. Their inner product is
+ \[
+ \langle f, g \rangle = \iint_{[0, 1]^2} f g^* \, d\mu d\nu.
+ \]
+ \end{definition}
+
+ \begin{definition}<2->
+ For a nice periodic function \(f(\mu, \nu)\): the numbers
+ \[
+ c_{m, n} = \langle f, B_{m, n} \rangle
+ \]
+ are the \emph{Fourier coefficients} or \emph{spectrum} of \(f\).
+ \end{definition}
+\end{frame}
+
+\begin{frame}{Fourier Series}
+ \begin{theorem}
+ For nice periodic functions:
+ \[
+ f(\mu, \nu) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}}
+ c_{m, n} B_{m, n} (\mu, \nu)
+ \]
+ where
+ \[
+ c_{m, n} = \langle f, B_{m, n} \rangle.
+ \]
+ \end{theorem}
+\end{frame}
+
+\begin{frame}{Why exponentials?}
+
+ \centering
+
+ {\huge\bfseries\itshape Why \(B_{m, n} = e^{i2\pi m\mu} e^{i2\pi n\nu}\)?}
+ \vspace{3em}
+
+ {\huge\bfseries\itshape Because
+ {\Huge \(\nabla^2\)}
+ }
+
+\end{frame}
+
+\begin{frame}{The Problem}
+ \begin{block}{Fourier's Problem}<1->
+ \[
+ \nabla^2 f(\mu, \nu)
+ = \frac{\partial^2 f}{\partial \mu^2} + \frac{\partial^2 f}{\partial \nu^2}
+ = \lambda f(\mu, \nu)
+ \]
+ \end{block}
+ \begin{alertblock}{Solution}<2->
+ Separation ansatz:
+ \[
+ f(\mu, \nu) = M(\mu) N(\nu)
+ \]
+ Resulting ODEs:
+ \begin{align*}
+ \frac{d^2 M}{d \mu^2} &= \kappa M(\mu), & \frac{d^2 N}{d \nu^2} &= (\lambda - \kappa) N(\nu)
+ \end{align*}
+ \end{alertblock}
+\end{frame}
+
+\section{The functions \(Y_{m, n}(\varphi, \vartheta)\)}
+
+\begin{frame}{Spherical Coordinates}
+ \begin{columns}
+ \begin{column}{.6\linewidth}
+ \includegraphics[width=\linewidth]{figures/spherical-coordinates}
+ \end{column}
+ \begin{column}{.4\linewidth}
+ \noindent
+ Variables
+ \begin{align*}
+ r &\in \mathbb{R}^+ \\
+ \vartheta &\in [0, \pi] \\
+ \varphi &\in [0, 2\pi)
+ \end{align*}
+ To cartesian
+ \begin{align*}
+ x &= r\cos\varphi \sin\vartheta \\
+ y &= r\sin\varphi\sin\vartheta \\
+ z &= r\cos\vartheta
+ \end{align*}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}{Spherical Laplacian}
+ \uncover<1->{
+ Cartesian Laplacian
+ \[
+ \nabla^2 \equiv \frac{\partial^2}{\partial \mu^2} + \frac{\partial^2}{\partial \nu^2}
+ \]
+ }
+
+ \uncover<2->{
+ Spherical Laplacian
+ \[
+ \nabla^2 \equiv
+ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)
+ + \frac{1}{r^2} \onslide<3-> \underbrace{ \onslide<2-> \left[
+ \frac{1}{\sin\vartheta} \frac{\partial}{\partial \vartheta}
+ \left( \sin\vartheta \frac{\partial}{\partial\vartheta} \right)
+ + \frac{1}{\sin^2 \vartheta} \frac{\partial^2}{\partial\varphi^2}
+ \right]
+ \onslide<3-> }_{\text{Surface Spherical Laplacian}~ \nabla^2_s} \onslide<2->
+ \]
+ }
+
+ \uncover<4->{
+ Surface Spherical Laplacian
+ \[
+ \nabla^2_s \equiv r^2 \nabla^2
+ - \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)
+ \]
+ }
+\end{frame}
+
+\bgroup
+\setbeamercolor{background canvas}{bg=black}
+\setbeamertemplate{navigation symbols}{}
+\begin{frame}{Geometrical Intuition}
+\end{frame}
+\egroup
+
+\begin{frame}{Where is \(\nabla^2_s\) useful?}
+ To do brain scans, apparently \cite{carvalhaes_surface_2015}
+ \only<1>{
+ \begin{figure}
+ \includegraphics[width=.8\linewidth, clip=100 0 0 0]{figures/eeg-photo}
+ \caption{Electroencephalogram (EEG). Image from Wikimedia \cite{baburov__2009}.}
+ \end{figure}
+ }
+ \only<2>{
+ \begin{figure} \centering
+ \includegraphics[width=\linewidth]{figures/surface-laplacian-eeg}
+ \caption{Surface Laplacian in EEG. Taken from \cite{ries_role_2013}.}
+ \end{figure}
+ }
+\end{frame}
+
+\begin{frame}{Brain Scans}
+ \begin{columns}
+ \begin{column}{.6\linewidth}
+ Electrodynamics
+ \begin{align*}
+ \nabla^2 \phi
+ &= \bm{\nabla \cdot} \bm{\nabla} \phi \qquad
+ \color{lightgray} \left( \phi = \int_\mathsf{A}^\mathsf{B} \vec{E} \bm{\cdot} d\vec{s} \right) \\
+ &= \bm{\nabla \cdot} \vec{E} \\
+ &\color{lightgray}= \int_{\Omega} (\bm{\nabla \cdot} \vec{E}) \bm{\cdot} d\vec{s}
+ = \oint_{\partial \Omega} \vec{E} \bm{\cdot} d\vec{s} \\
+ &= \frac{\rho}{\varepsilon}
+ \end{align*}
+ So over the scalp
+ \[
+ \nabla^2_s \phi
+ = \frac{\rho_s}{\varepsilon}
+ = \text{Current flow in the brain}
+ \]
+ \end{column}
+ \begin{column}{.4\linewidth}
+ \centering
+ \includegraphics[width=\linewidth]{figures/flux}
+ \nocite{maschen_divergence_2013}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}{New Hard Problem}
+ \begin{block}{The Problem}<1->
+ \only<1>{
+ \[
+ \nabla^2_s f(\varphi, \vartheta) = \lambda f(\varphi, \vartheta)
+ \]
+ }
+ \only<2->{
+ \[
+ \frac{1}{\sin\vartheta} \frac{\partial}{\partial \vartheta}
+ \left( \sin\vartheta \frac{\partial f}{\partial\vartheta} \right)
+ + \frac{1}{\sin^2 \vartheta} \frac{\partial^2 f}{\partial\varphi^2}
+ = \lambda f(\varphi, \vartheta)
+ \]
+ }
+ \end{block}
+ \begin{alertblock}{Idea}<3->
+ Separation ansatz:
+ \[
+ f(\varphi, \vartheta) = \Phi(\varphi) \Theta(\vartheta)
+ \]
+ From the ``easy'' part:
+ \[
+ \frac{d^2\Phi}{d\varphi^2} = \kappa \Phi(\varphi)
+ \implies \Phi(\varphi) = e^{im\varphi},
+ \quad m \in \mathbb{Z}
+ \]
+ \end{alertblock}
+\end{frame}
+
+\begin{frame}{Associated Legendre Differential Equation}
+ \begin{alertblock}{Separation (cont.)}<1->
+ The hard part is the ODE for \(\Theta(\vartheta)\):
+ \[
+ \sin^2\vartheta \frac{d^2 \Theta}{d (\cos\vartheta)^2} - 2\cos\theta \frac{d \Theta}{d \cos\vartheta}
+ + \left[ n(n+1) - \frac{m^2}{\sin^2 \vartheta} \right] \Theta(\cos\vartheta) = 0
+ \]
+ \end{alertblock}
+
+ \uncover<2->{
+ Substituting \(x = \cos\vartheta\) and \(y = \Theta\):
+ }
+
+ \begin{definition}<2->[Associated Legendre Differential Equation]
+ \[
+ \left( 1 - x^2 \right) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx}
+ + \left[ n(n+1) - \frac{m^2}{1 - x^2} \right] y(x) = 0
+ \]
+ \end{definition}
+\end{frame}
+
+\begin{frame}{Legendre Polynomials}
+ \begin{definition}[Legendre Polynomials]
+ The polynomials
+ \begin{align*}
+ P_n(x)
+ &= \sum_{k=0}^{\lfloor n/2 \rfloor}
+ \frac{(-1)^k (2n-2k)!}{2^n k! (n-k)!(n-2k)!} x^{n-2k} \\[1em]
+ &= {}_2F_1 \left( \begin{matrix}
+ n + 1, & -n \\ \multicolumn{2}{c}{1}
+ \end{matrix} ; \frac{1 - x}{2} \right) \\[1em]
+ &= \frac{1}{n!2^n}\frac{d^n}{dx^n}(x^2-1)^n
+ \end{align*}
+ are a solution to the associated Legendre differential equation when \(m = 0\).
+ \end{definition}
+\end{frame}
+
+\begin{frame}
+ \centering
+ \includegraphics[width=\linewidth]{figures/legendre-polynomials}
+\end{frame}
+
+\begin{frame}{Associated Legendre Polynomials}
+ \begin{lemma}
+ For \(x \in [-1, 1]\) the polynomials
+ \[
+ P_{m, n} (x) = \left( 1 - x^2 \right)^{m/2} \frac{d^{m}}{dx^{m}} P_n (x)
+ \]
+ solve the associated Legendre differential equation.
+ \end{lemma}
+
+ \begin{alertblock}{Observation}<2->
+ If \(m > n\) then \(P_{m, n}(x) = 0\) for all \(x\).
+ \end{alertblock}
+\end{frame}
+
+\begin{frame}
+ \centering
+ \includegraphics[width=\linewidth]{figures/associated-legendre-polynomials}
+\end{frame}
+
+
+\begin{frame}{Putting it back together}
+ \begin{block}{The Problem}
+ \[
+ \nabla^2_s f(\varphi, \vartheta) = \lambda f(\varphi, \vartheta)
+ \]
+ \end{block}
+ \begin{alertblock}{Current solution}
+ For \(m \in \mathbb{Z}\) and \(m < n\):
+ \[
+ \tilde{Y}_{m, n}(\varphi, \vartheta)
+ = \Phi(\varphi) \Theta(\vartheta)
+ = e^{im\varphi} P_{m, n}(\cos\vartheta)
+ \]
+ \end{alertblock}
+\end{frame}
+
+\bgroup
+\setbeamercolor{background canvas}{bg=black}
+\setbeamertemplate{navigation symbols}{}
+\begin{frame}{Intuition of conditions for \(m\) and \(n\)}
+\end{frame}
+\egroup
+
+\section{Fourier on \(S^2\)}
+
+\begin{frame}{Basis functions?}
+ The functions \(\tilde{Y}_{m, n}\) span the space of nice functions \(S^2 \to \mathbb{C}\).
+
+ \begin{definition}<2->
+ The inner product of nice functions \(f(\varphi, \vartheta)\) and \(g(\varphi, \vartheta)\) from \(S^2\) to \(\mathbb{C}\) is
+ \[
+ \langle f, g \rangle
+ = \iint_{S^2} f g^* \, d\Omega
+ \uncover<3->{
+ = \int\limits_0^{2\pi} \int\limits_0^{\pi}
+ f(\varphi, \vartheta) g^*(\varphi, \vartheta)
+ \sin\vartheta \, d\vartheta d\varphi
+ }
+ \]
+ \end{definition}
+\end{frame}
+
+\begin{frame}{Orthonormality}
+ \begin{definition}<1->
+ A set of basis functions are \emph{orthonormal} if
+ \[
+ \langle B_{m, n}, B_{m', n'} \rangle = \begin{cases}
+ 1 & m = m' \wedge n = n' \\
+ 0 & \text{else}
+ \end{cases}
+ \]
+ \end{definition}
+
+ \begin{alertblock}{Problem}<2->
+ \[
+ \langle \tilde{Y}_{m, n}, \tilde{Y}_{m', n'} \rangle = \begin{cases} \displaystyle
+ \frac{4 \pi}{2n+1} \frac{(n+m)!}{(n-m)!} & m = m' \wedge n = n' \\
+ 0 & \text{else}
+ \end{cases}
+ \]
+ \end{alertblock}
+\end{frame}
+
+\begin{frame}{Spherical Harmonics}
+ \begin{definition}<1->
+ The orthonormal spherical harmonics are
+ \[
+ Y_{m, n}(\varphi, \vartheta) = N_{m, n} e^{im\varphi} P_{m, n}(\cos\vartheta)
+ \]
+ where the normalisation constant
+ % FIXME: (-1)^m
+ \[
+ N_{m, n} = \sqrt{\frac{2n+1}{4 \pi} \frac{(n-m)!}{(n+m)!}}
+ \]
+ \end{definition}
+ \begin{alertblock}{Fixed}<1->
+ \[
+ \langle Y_{m, n}, Y_{m', n'} \rangle = \begin{cases}
+ 1 & m = m' \wedge n = n' \\
+ 0 & \text{else}
+ \end{cases}
+ \]
+ \end{alertblock}
+\end{frame}
+
+\begin{frame}{Fourier Series}
+ \begin{theorem}
+ For nice periodic functions on \(S^2\):
+ \[
+ f(\varphi, \vartheta) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}}
+ c_{m, n} Y_{m, n} (\varphi, \vartheta)
+ \]
+ where
+ \[
+ c_{m, n} = \langle f, Y_{m, n} \rangle.
+ \]
+ \end{theorem}
+\end{frame}
+
+\section{Quantum Mechanics}
+
+\begin{frame}{Linear and Rotational Kinetic Energy}
+ \begin{columns}
+ \begin{column}{.5\linewidth}
+ \begin{block}{Momentum and KE}<1->
+ \[
+ \vec{p} = m \vec{v},
+ \quad
+ E_k = \frac{\vec{p}^2}{2m}
+ \]
+ \end{block}
+ \begin{alertblock}{QM Formulation}<3->
+ \[
+ \vec{\hat{p}} = -i\hbar \bm{\nabla},
+ \quad
+ \hat{E}_k = -\frac{\hbar^2}{2m} \nabla^2
+ \]
+ \end{alertblock}
+ \end{column}
+ \begin{column}{.5\linewidth}<2->
+ \begin{block}{Angular M. and KE}
+ \[
+ \vec{L} = \vec{r}\bm{\times}{\vec{p}},
+ \quad
+ E_{k, a} = \frac{\vec{L}^2}{2m r^2}
+ \]
+ \end{block}
+ \begin{alertblock}{QM Formulation}<4->
+ Pretty long derivation yields:
+ \begin{align*}
+ % \hat{L}_z &= -i \hbar \frac{\partial}{\partial \varphi}, \\[1em]
+ \hat{E}_{k, a} &= -\frac{\hbar^2}{2mr^2} \nabla^2_s
+ \end{align*}
+ \end{alertblock}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+
+\bgroup
+\setbeamercolor{background canvas}{bg=black}
+\setbeamertemplate{navigation symbols}{}
+\begin{frame}{Intuition for the Operators}
+\end{frame}
+\egroup
+
+\begin{frame}{Schrödinger Equation}
+ \begin{block}{Time independent SE}
+ \[
+ % hamiltonina
+ \only<1>{
+ \mathrm{\hat{\mathcal{H}}} | \Psi \rangle = E | \Psi \rangle
+ }
+ % KE + U
+ \only<2>{
+ \left(
+ \hat{E}_k + U
+ \right) | \Psi \rangle = E | \Psi \rangle
+ }
+ % KE with p
+ \only<3>{
+ \left(
+ \frac{\vec{\hat{p}}^2}{2m} + U
+ \right) | \Psi \rangle = E | \Psi \rangle
+ }
+ % KE with p as 1D derivative
+ \only<4>{
+ \text{Meili} \qquad
+ \left[
+ - \frac{\hbar^2}{2m} \frac{d^2}{d x^2} + U(x)
+ \right] \Psi(x) = E \Psi(x)
+ }
+ % KE with p as 3D derivative
+ \only<5>{
+ \text{3D} \qquad
+ \left[
+ - \frac{\hbar^2}{2m} \nabla^2 + U(\vec{x})
+ \right] \Psi(\vec{x}) = E \Psi(\vec{x})
+ }
+ % Decompose laplacian
+ \only<6>{
+ \left\{
+ - \frac{\hbar^2}{2m} \frac{1}{r^2} \left[
+ \nabla^2_s - \frac{\partial}{\partial r} \left(
+ r^2 \frac{\partial}{\partial r}
+ \right)
+ \right] + U(\vec{r})
+ \right\} \Psi(\vec{r}) = E \Psi(\vec{r})
+ }
+ % rewrite using L
+ \only<7>{
+ \left[
+ \frac{\vec{\hat{L}}^2}{2mr^2}
+ + \frac{1}{r^2} \frac{\partial}{\partial r} \left(
+ r^2 \frac{\partial}{\partial r}
+ \right)
+ + U(\vec{r})
+ \right] \Psi(\vec{r}) = E \Psi(\vec{r})
+ }
+ % rewrite using E_ka
+ \only<8>{
+ \left[
+ \hat{E}_{k,a}
+ + \frac{1}{r^2} \frac{\partial}{\partial r} \left(
+ r^2 \frac{\partial}{\partial r}
+ \right)
+ + U(\vec{r})
+ \right] \Psi(\vec{r}) = E \Psi(\vec{r})
+ }
+ % What is KE
+ \only<9>{
+ \Bigg[
+ \underbrace{\hat{E}_{k,a}
+ + \frac{1}{r^2} \frac{\partial}{\partial r} \left(
+ r^2 \frac{\partial}{\partial r}
+ \right)}_\text{Kinetic Energy}
+ + U(\vec{r})
+ \Bigg] \Psi(\vec{r}) = E \Psi(\vec{r})
+ }
+ % Introduce E_kr
+ \only<10>{
+ \Bigg[
+ \hat{E}_{k,a}
+ + \underbrace{\frac{1}{r^2} \frac{\partial}{\partial r} \left(
+ r^2 \frac{\partial}{\partial r}
+ \right)}_{\text{Radial KE } \hat{E}_{k, r}}
+ + U(\vec{r})
+ \Bigg] \Psi(\vec{r}) = E \Psi(\vec{r})
+ }
+ \only<11->{
+ \left\{
+ \hat{E}_{k,a} + \hat{E}_{k,r} + U(\vec{r})
+ \right\} \Psi(\vec{r}) = E \Psi(\vec{r})
+ }
+ \]
+ \end{block}
+ \begin{columns}
+ \begin{column}{.6\linewidth}
+ \Large
+ \uncover<11->{
+ \Large
+ \textit{But why?} \\[2em]
+ }
+
+ \uncover<12->{
+ \bfseries
+ Hydrogen atom has radial symmetry!
+ }
+ \end{column}
+ \begin{column}{.35\linewidth}
+ \uncover<11->{
+ \includegraphics[width=\linewidth]{figures/hydrogen}
+ \nocite{depiep_electron_2013}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}{Electron Orbitals}
+\end{frame}
+
+% \section{Other applications}
+
+\begin{frame}{Bibliography}
+ \renewcommand*{\bibfont}{\tiny}
+ \printbibliography
+\end{frame}
+
+\end{document}
+
+% vim:et:ts=2:sw=2:wrap:nolinebreak:
diff --git a/presentation/Makefile b/presentation/Makefile
new file mode 100644
index 0000000..8427cb0
--- /dev/null
+++ b/presentation/Makefile
@@ -0,0 +1,19 @@
+TEX := xelatex
+TEXARGS := --output-directory=build --halt-on-error
+
+DOCNAME := FourierOnS2
+SOURCES := $(DOCNAME).tex
+
+include tex/Makefile.inc
+
+.PHONY: clean
+all: build/$(DOCNAME).pdf
+
+clean:
+ @rm -rfv build
+
+build/$(DOCNAME).pdf : $(SOURCES)
+ mkdir -p build
+ $(TEX) $(TEXARGS) $<
+ $(TEX) $(TEXARGS) $<
+
diff --git a/presentation/beamercolorthemehsr.sty b/presentation/beamercolorthemehsr.sty
new file mode 100644
index 0000000..e167ab0
--- /dev/null
+++ b/presentation/beamercolorthemehsr.sty
@@ -0,0 +1,43 @@
+
+\setbeamercolor{alerted text}{fg=hsr-mauve}
+\setbeamercolor{background canvas}{bg=white}
+
+% blocks body
+\setbeamercolor{block body}{bg=hsr-lightgrey20}
+\setbeamercolor{block body example}{bg=hsr-lightgrey20}
+\setbeamercolor{block body alerted}{bg=hsr-lightgrey20}
+
+% block titles
+\setbeamercolor{block title}{bg=hsr-blue, fg=white}
+\setbeamercolor{block title alerted}{bg=hsr-mauve80, fg=white}
+\setbeamercolor{block title example}{bg=hsr-lakegreen, fg=white}
+
+% title and text
+\setbeamercolor{title}{bg=white, fg=hsr-black}
+\setbeamercolor{titlelike}{bg=hsr-blue, fg=white}
+
+\setbeamercolor{frametitle}{fg=white}
+\setbeamercolor{item projected}{fg=white}
+
+\setbeamercolor{normal text}{fg=hsr-black}
+
+\setbeamercolor{palette sidebar primary}{use=normal text,fg=normal text.fg}
+\setbeamercolor{palette sidebar quaternary}{use=structure,fg=structure.fg}
+\setbeamercolor{palette sidebar secondary}{use=structure,fg=structure.fg}
+\setbeamercolor{palette sidebar tertiary}{use=normal text,fg=normal text.fg}
+
+\setbeamercolor{fine separation line}{}
+\setbeamercolor{separation line}{}
+
+% structures (bullet points etc)
+\setbeamercolor{structure}{bg=hsr-lightgrey20, fg=hsr-blue}
+
+% sidebar stuff
+\setbeamercolor{sidebar}{bg=hsr-blue}
+\setbeamercolor{sidebar}{parent=palette primary}
+
+\setbeamercolor{section in sidebar}{fg=hsr-black}
+\setbeamercolor{section in sidebar shaded}{fg=hsr-black40}
+\setbeamercolor{subsection in sidebar}{fg=hsr-black}
+\setbeamercolor{subsection in sidebar shaded}{fg=hsr-black40}
+
diff --git a/presentation/beamerthemehsr.log b/presentation/beamerthemehsr.log
new file mode 100644
index 0000000..a469eb0
--- /dev/null
+++ b/presentation/beamerthemehsr.log
@@ -0,0 +1,28 @@
+This is XeTeX, Version 3.141592653-2.6-0.999994 (TeX Live 2022) (preloaded format=xelatex 2022.4.19) 16 MAY 2022 19:11
+entering extended mode
+ restricted \write18 enabled.
+ file:line:error style messages enabled.
+ %&-line parsing enabled.
+**beamerthemehsr.sty
+(./beamerthemehsr.sty
+LaTeX2e <2021-11-15> patch level 1
+L3 programming layer <2022-02-24>
+./beamerthemehsr.sty:1: Undefined control sequence.
+l.1 \mode
+ <presentation>
+?
+./beamerthemehsr.sty:1: Emergency stop.
+l.1
+
+End of file on the terminal!
+
+
+Here is how much of TeX's memory you used:
+ 16 strings out of 478142
+ 329 string characters out of 5851554
+ 290467 words of memory out of 5000000
+ 20742 multiletter control sequences out of 15000+600000
+ 469259 words of font info for 28 fonts, out of 8000000 for 9000
+ 14 hyphenation exceptions out of 8191
+ 12i,0n,12p,69b,10s stack positions out of 10000i,1000n,20000p,200000b,200000s
+No pages of output.
diff --git a/presentation/beamerthemehsr.sty b/presentation/beamerthemehsr.sty
new file mode 100644
index 0000000..71af531
--- /dev/null
+++ b/presentation/beamerthemehsr.sty
@@ -0,0 +1,77 @@
+\mode<presentation>
+
+\RequirePackage{graphicx}
+\RequirePackage{tikz}
+\RequirePackage{xcolor}
+
+% define HSR Theme Colors V3.0
+%% blue
+\definecolor{hsr-blue}{HTML}{0065A3}
+\definecolor{hsr-blue80}{HTML}{3384B5}
+\definecolor{hsr-blue60}{HTML}{66A3C8}
+\definecolor{hsr-blue40}{HTML}{99C1DA}
+\definecolor{hsr-blue20}{HTML}{CCE0ED}
+
+%% mauve / hematite
+\definecolor{hsr-mauve}{HTML}{6E1C50}
+\definecolor{hsr-mauve80}{HTML}{8B4973}
+\definecolor{hsr-mauve60}{HTML}{A87796}
+\definecolor{hsr-mauve40}{HTML}{C5A4B9}
+\definecolor{hsr-mauve20}{HTML}{E2D2DC}
+
+%% lakegreen
+\definecolor{hsr-lakegreen}{HTML}{548C86}
+\definecolor{hsr-lakegreen80}{HTML}{76A39E}
+\definecolor{hsr-lakegreen60}{HTML}{98BAB6}
+\definecolor{hsr-lakegreen40}{HTML}{BBD1CF}
+\definecolor{hsr-lakegreen20}{HTML}{DDE8E7}
+
+%% reed
+\definecolor{hsr-reed}{HTML}{7B6951}
+\definecolor{hsr-reed80}{HTML}{958774}
+\definecolor{hsr-reed60}{HTML}{B0A597}
+\definecolor{hsr-reed40}{HTML}{CAC3B9}
+\definecolor{hsr-reed20}{HTML}{E5E1DC}
+
+%% petrol
+\definecolor{hsr-petrol}{HTML}{00738D}
+\definecolor{hsr-petrol80}{HTML}{338FA4}
+\definecolor{hsr-petrol60}{HTML}{66ABBB}
+\definecolor{hsr-petrol40}{HTML}{99C7D1}
+\definecolor{hsr-petrol20}{HTML}{CCE3E8}
+
+%% basswood
+\definecolor{hsr-basswood}{HTML}{BABD5D}
+\definecolor{hsr-basswood80}{HTML}{C8CA7D}
+\definecolor{hsr-basswood60}{HTML}{D6D79E}
+\definecolor{hsr-basswood40}{HTML}{E3E5BE}
+\definecolor{hsr-basswood20}{HTML}{F1F2DF}
+
+%% lightgrey
+\definecolor{hsr-lightgrey}{HTML}{C6C7C8}
+\definecolor{hsr-lightgrey80}{HTML}{D1D2D3}
+\definecolor{hsr-lightgrey60}{HTML}{DDDDDE}
+\definecolor{hsr-lightgrey40}{HTML}{E8E8E9}
+\definecolor{hsr-lightgrey20}{HTML}{F4F4F4}
+
+%% black
+\definecolor{hsr-black}{HTML}{1A171B}
+\definecolor{hsr-black80}{HTML}{484549}
+\definecolor{hsr-black60}{HTML}{767476}
+\definecolor{hsr-black40}{HTML}{A4A2A4}
+\definecolor{hsr-black20}{HTML}{D1D1D1}
+
+\useinnertheme{rectangles}
+\useoutertheme{default}
+\usecolortheme{hsr}
+
+\setbeamerfont{title}{size=\Huge}
+
+\setbeamerfont{frametitle}{size=\Large}
+
+\setbeamertemplate{caption}[numbered]
+\setbeamertemplate{navigation symbols}{%
+ \insertslidenavigationsymbol%
+ \insertsectionnavigationsymbol%
+}
+\mode<all>
diff --git a/presentation/figures/associated-legendre-polynomials.pdf b/presentation/figures/associated-legendre-polynomials.pdf
new file mode 100644
index 0000000..97ad447
--- /dev/null
+++ b/presentation/figures/associated-legendre-polynomials.pdf
Binary files differ
diff --git a/presentation/figures/buchcover.pdf b/presentation/figures/buchcover.pdf
new file mode 100644
index 0000000..0285183
--- /dev/null
+++ b/presentation/figures/buchcover.pdf
Binary files differ
diff --git a/presentation/figures/eeg-photo.jpg b/presentation/figures/eeg-photo.jpg
new file mode 100644
index 0000000..cb06d80
--- /dev/null
+++ b/presentation/figures/eeg-photo.jpg
Binary files differ
diff --git a/presentation/figures/flat-basis-functions.pdf b/presentation/figures/flat-basis-functions.pdf
new file mode 100644
index 0000000..8092e26
--- /dev/null
+++ b/presentation/figures/flat-basis-functions.pdf
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diff --git a/presentation/figures/flux.pdf b/presentation/figures/flux.pdf
new file mode 100644
index 0000000..6a87288
--- /dev/null
+++ b/presentation/figures/flux.pdf
Binary files differ
diff --git a/presentation/figures/flux.svg b/presentation/figures/flux.svg
new file mode 100644
index 0000000..cee01a7
--- /dev/null
+++ b/presentation/figures/flux.svg
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diff --git a/presentation/figures/hydrogen.pdf b/presentation/figures/hydrogen.pdf
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diff --git a/presentation/figures/hydrogen.svg b/presentation/figures/hydrogen.svg
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+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!DOCTYPE svg>
+<svg version="1.1" baseProfile="full" xmlns:ev="http://www.w3.org/2001/xml-events" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" preserveAspectRatio="xMidYMid meet" zoomAndPan="magnify" id="Electron shell Hydrogen" viewBox="-400 -400 800 800" width="800" height="800">
+<title id="title">Electron shells for element: Z=001 H (Hydrogen): 1</title>
+<desc id="element_description">001 H (Hydrogen) Legend_color:#e7ff8f</desc>
+<desc id="element_electron_configuration">1</desc>
+
+<defs>
+<circle id="electron" fill="#77868b" stroke="none" r="30"/>
+</defs>
+
+<circle fill="#e7ff8f" r="200" stroke="#343434" stroke-width="10"/>
+<g font-weight="bold" font="DejaVu Sans">
+<text x="0" y="50" font-size="150" text-anchor="middle">H</text>
+</g>
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+</g>
+</svg> \ No newline at end of file
diff --git a/presentation/figures/legendre-polynomials.pdf b/presentation/figures/legendre-polynomials.pdf
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diff --git a/presentation/figures/spherical-coordinates.pdf b/presentation/figures/spherical-coordinates.pdf
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diff --git a/presentation/figures/spherical-coordinates.tex b/presentation/figures/spherical-coordinates.tex
new file mode 100644
index 0000000..3a45385
--- /dev/null
+++ b/presentation/figures/spherical-coordinates.tex
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+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{bm}
+\usepackage{lmodern}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+
+\tdplotsetmaincoords{60}{130}
+\pgfmathsetmacro{\l}{2}
+
+\begin{tikzpicture}[
+ >=latex,
+ tdplot_main_coords,
+ dot/.style = {
+ black, fill = black, circle,
+ outer sep = 0, inner sep = 0,
+ minimum size = .8mm
+ },
+ round/.style = {
+ draw = orange, thick, circle,
+ minimum size = 1mm,
+ inner sep = 0pt, outer sep = 0pt,
+ },
+ cross/.style = {
+ cross out, draw = magenta, thick,
+ minimum size = 1mm,
+ inner sep = 0pt, outer sep = 0pt
+ },
+ ]
+
+ % origin
+ \coordinate (O) at (0,0,0);
+
+ % poles
+ \coordinate (NP) at (0,0,\l);
+ \coordinate (SP) at (0,0,-\l);
+
+ % \draw (SP) node[dot, gray] {};
+ % \draw (NP) node[dot, gray] {};
+
+ % gray unit circle
+ \tdplotdrawarc[gray]{(O)}{\l}{0}{360}{}{};
+ \draw[gray, dashed] (-\l, 0, 0) to (\l, 0, 0);
+ \draw[gray, dashed] (0, -\l, 0) to (0, \l, 0);
+
+ % axis
+ \draw[->] (O) -- ++(1.25*\l,0,0) node[left] {\(\mathbf{\hat{x}}\)};
+ \draw[->] (O) -- ++(0,1.25*\l,0) node[right] {\(\mathbf{\hat{y}}\)};
+ \draw[->] (O) -- ++(0,0,1.25*\l) node[above] {\(\mathbf{\hat{z}}\)};
+
+ % meridians
+ \foreach \phi in {0, 30, 60, ..., 150}{
+ \tdplotsetrotatedcoords{\phi}{90}{0};
+ \tdplotdrawarc[lightgray, densely dotted, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
+ }
+
+ % dot above and its projection
+ \pgfmathsetmacro{\phi}{120}
+ \pgfmathsetmacro{\theta}{40}
+
+ \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l}
+ \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l}
+ \pgfmathsetmacro{\pz}{cos(\theta)*\l})
+
+ % point A
+ \coordinate (A) at (\px,\py,\pz);
+ \coordinate (Ap) at (\px,\py, 0);
+
+ % lines
+ \draw[red!80!black, ->] (O) -- (A);
+ \draw[red!80!black, densely dashed] (O) -- (Ap) -- (A)
+ node[above right] {\(\mathbf{\hat{r}}\)};
+
+ % arcs
+ \tdplotdrawarc[blue!80!black, ->]{(O)}{.8\l}{0}{\phi}{}{};
+ \node[below right, blue!80!black] at (.8\l,0,0) {\(\bm{\hat{\varphi}}\)};
+
+ \tdplotsetrotatedcoords{\phi-90}{-90}{0};
+ \tdplotdrawarc[blue!80!black, ->, tdplot_rotated_coords]{(O)}{.95\l}{0}{\theta}{}{};
+ \node[above right = 1mm, blue!80!black] at (0,0,.8\l) {\(\bm{\hat{\vartheta}}\)};
+
+
+ % dots
+ \draw (O) node[dot] {};
+ \draw (A) node[dot, fill = red!80!black] {};
+
+\end{tikzpicture}
+\end{document}
+% vim:ts=2 sw=2 et:
diff --git a/presentation/figures/surface-laplacian-eeg.pdf b/presentation/figures/surface-laplacian-eeg.pdf
new file mode 100644
index 0000000..f066a30
--- /dev/null
+++ b/presentation/figures/surface-laplacian-eeg.pdf
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