From e519158c792228454bb2361b6a2b5d1e82ecaaee Mon Sep 17 00:00:00 2001 From: Lars Bogner Date: Tue, 23 Sep 2025 10:09:09 +0200 Subject: [PATCH] Finish proj 2 --- projects/project2/include/main.tex | 119 ++++++++++++++++++++++++ projects/project2/include/mainNotes.bib | 2 + 2 files changed, 121 insertions(+) create mode 100644 projects/project2/include/main.tex create mode 100644 projects/project2/include/mainNotes.bib diff --git a/projects/project2/include/main.tex b/projects/project2/include/main.tex new file mode 100644 index 0000000..3a82feb --- /dev/null +++ b/projects/project2/include/main.tex @@ -0,0 +1,119 @@ +\documentclass[english,notitlepage]{revtex4-1} % defines the basic parameters of the document +%For preview: skriv i terminal: latexmk -pdf -pvc filnavn +\let\tablenum\relax % to avoid clash with siunitx + + +% if you want a single-column, remove reprint + +% allows special characters (including æøå) +\usepackage[utf8]{inputenc} +%\usepackage[english]{babel} + +%% note that you may need to download some of these packages manually, it depends on your setup. +%% I recommend downloading TeXMaker, because it includes a large library of the most common packages. + +\usepackage{amssymb} % mathematical symbols (physics imports amsmath) +\usepackage{amsmath} +\usepackage{graphicx} % include graphics such as plots +\usepackage{xcolor} % set colors +\usepackage{hyperref} % automagic cross-referencing (this is GODLIKE) +\usepackage{listings} % display code +\usepackage{subfigure} % imports a lot of cool and useful figure commands +\usepackage{float} +%\usepackage[section]{placeins} +\usepackage{algorithm} +\usepackage[noend]{algpseudocode} +\usepackage{subfigure} +\usepackage{tikz} +\usepackage{siunitx} +\usetikzlibrary{quantikz} +% defines the color of hyperref objects +% Blending two colors: blue!80!black = 80% blue and 20% black +\hypersetup{ % this is just my personal choice, feel free to change things + colorlinks, + linkcolor={red!50!black}, + citecolor={blue!50!black}, + urlcolor={blue!80!black}} + +%% Defines the style of the programming listing +%% This is actually my personal template, go ahead and change stuff if you want + + + +%% USEFUL LINKS: +%% +%% UiO LaTeX guides: https://www.mn.uio.no/ifi/tjenester/it/hjelp/latex/ +%% mathematics: https://en.wikibooks.org/wiki/LaTeX/Mathematics + +%% PHYSICS ! https://mirror.hmc.edu/ctan/macros/latex/contrib/physics/physics.pdf + +%% the basics of Tikz: https://en.wikibooks.org/wiki/LaTeX/PGF/Tikz +%% all the colors!: https://en.wikibooks.org/wiki/LaTeX/Colors +%% how to draw tables: https://en.wikibooks.org/wiki/LaTeX/Tables +%% code listing styles: https://en.wikibooks.org/wiki/LaTeX/Source_Code_Listings +%% \includegraphics https://en.wikibooks.org/wiki/LaTeX/Importing_Graphics +%% learn more about figures https://en.wikibooks.org/wiki/LaTeX/Floats,_Figures_and_Captions +%% automagic bibliography: https://en.wikibooks.org/wiki/LaTeX/Bibliography_Management (this one is kinda difficult the first time) +%% REVTeX Guide: http://www.physics.csbsju.edu/370/papers/Journal_Style_Manuals/auguide4-1.pdf +%% +%% (this document is of class "revtex4-1", the REVTeX Guide explains how the class works) + + +%% CREATING THE .pdf FILE USING LINUX IN THE TERMINAL +%% +%% [terminal]$ pdflatex template.tex +%% +%% Run the command twice, always. +%% If you want to use \footnote, you need to run these commands (IN THIS SPECIFIC ORDER) +%% +%% [terminal]$ pdflatex template.tex +%% [terminal]$ bibtex template +%% [terminal]$ pdflatex template.tex +%% [terminal]$ pdflatex template.tex +%% +%% Don't ask me why, I don't know. + +\newcommand{\tderiv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} % defines a command for total derivatives +\newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} % defines a command for partial derivatives +\newcommand{\ptderiv}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d} #2^{#3}}} % defines a command for total derivatives of arbitrary power +\newcommand{\ppderiv}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}} % defines a command for partial derivatives of arbitrary power + +\begin{document} + +\title{Solutions to Project 2} % self-explanatory +\author{Lars Bogner} % self-explanatory +\date{\today} % self-explanatory +\noaffiliation % ignore this, but keep it. + + +\maketitle + +\textit{Code hosted at \url{https://github.uio.no/larsbog/FYS4150}} + +\section*{Problem 1} +Using the rescaling of the positions $\hat x \equiv \frac{x}{L}$ we define the following relations: +\begin{equation} + u_x \equiv u, \quad u_{\hat x} \equiv \hat u, \quad \text{with} \quad u(x) = \hat u(\hat x) = u (L \hat x). +\end{equation} +Now we can rewrite the equation as +\begin{equation} + \begin{aligned} + \gamma \ptderiv{u(x)}{x}{2} &= - F u(x)\\ + \gamma \ptderiv{u(L \hat x)}{(L \hat x)}{2} &= - F u(L \hat x). + \end{aligned} +\end{equation} +In this step we simply replaced $x$ by the equivalent expression $L \hat x$. Now we can use the chain rule to rewrite the left hand side: +\begin{equation} + \gamma \ppderiv{u(L \hat x)}{\hat x}{2} \underbrace{\ppderiv{\hat x}{(L \hat x)}{2}}_{=L^{-2}} = - F u(L \hat x). +\end{equation} +Rearranging the factors gives the final result: +\begin{equation} + \ptderiv{\hat u (\hat x)}{\hat x}{2} = - \frac{F L^2}{\gamma} \hat u(\hat x) \equiv - \lambda \hat u(\hat x). +\end{equation} + +\section*{Problem 2} + + +\bibliography{include/citations} +\end{document} + \ No newline at end of file diff --git a/projects/project2/include/mainNotes.bib b/projects/project2/include/mainNotes.bib new file mode 100644 index 0000000..f275e40 --- /dev/null +++ b/projects/project2/include/mainNotes.bib @@ -0,0 +1,2 @@ +@CONTROL{REVTEX41Control} +@CONTROL{apsrev41Control,author="08",editor="1",pages="0",title="",year="1"}