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Mqf%0e$-PJY9~e;Lvj6}9 literal 0 HcmV?d00001 diff --git a/semester3/analysis-ii-rb/main.tex b/semester3/analysis-ii-rb/main.tex index e69de29..b4b19ee 100644 --- a/semester3/analysis-ii-rb/main.tex +++ b/semester3/analysis-ii-rb/main.tex @@ -0,0 +1,15 @@ +\documentclass[a4paper,10pt]{article} +\usepackage[landscape, left=0.75cm, top=1cm, right=0.75cm, bottom=1.5cm, footskip=15pt]{geometry} +\input{util/setup.tex} +\input{util/helpers.tex} + +\title{Analysis I} +\author{Robin Bacher} +\date{FS 2025} + +\begin{document} + +\section{Differential Equations} +\input{parts/01_diffeq.tex} + +\end{document} diff --git a/semester3/analysis-ii-rb/parts/01_diffeq.tex b/semester3/analysis-ii-rb/parts/01_diffeq.tex new file mode 100644 index 0000000..99526d2 --- /dev/null +++ b/semester3/analysis-ii-rb/parts/01_diffeq.tex @@ -0,0 +1,117 @@ +\definition \textbf{Differential Equation} (DE)\\ +Equation relating unknown $f$ to derivatives $f^{(i)}$ at \textit{same} $x$. + +\definition \textbf{Ordinary Differential Equation} (ODE)\\ +DE s.t. $f: I \to \R$ is in one variable. + +\definition \textbf{Partial Differential Equation} (PDE)\\ +DE s.t. $f: I^d \to \R$ is in multiple variables. + +\notation $f^{(i)}$ or $y^{(i)}$ instead of $f^{(i)}(x)$ for brevity. + +\definition \textbf{Order} $\ \ord(F) := \underset{i \geq 0}{\text{max}}\{ i \sep f^{(i)} \in F,\ f^{(i)} \neq 0 \}$ + +\remark Any $F$ s.t. $\ord(F) \geq 2$ can be reduced to $\ord(F') = 1$, but using functions of higher dimensions. + +\begin{subbox}{Solutions to ODEs} + \smalltext{$\forall F: \R^2 \to \R$ s.t. $F$ is cont. diff. and $x_0,y_0 \in \R$:} + \begin{align*} + & \exists f: I \to \R \\ + & \text{s.t. } \forall x \in I: f'(x) = F(x, f(x)) \text{ and } f(x_0) = y_0 + \end{align*} + \smalltext{s.t. $I$ is open and maximal.} +\end{subbox} +\subtext{Intuition: Solutions always exist (locally!) for \textit{nice enough} equations.} + +\subsection{Linear Differential Equations} + +\definition \textbf{Linear Differential Equation} (LDE)\\ +$$ +y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b +$$ +\subtext{ + $I \subset \R$ is open$,\quad k \geq 1,\quad \forall i < k: a_i: I \to \C$ +} + +\definition Homogeneity of LDEs\\ +\begin{tabular}{ll} + \textbf{Homogeneous} & $\iffdef b = 0$\\ + \textbf{Inhomogeneous} & $\iffdef b \neq 0$ +\end{tabular} + +\remark $D(y) := y^{(k)} + \ldots + a_0y$ is a linear operation: +$$ +D(z_1f_1 + z_2f_2) = z_1D(f_1) + z_2D(f_2) +$$ +\subtext{ + $\forall z_1,z_2 \in \C,\quad f_1,f_2\ k$-times differentiable: +} + +\definition \textbf{Homogeneous Solution Space}\\ +$\S(F) := \{ f: I \to \C \sep f \text{ solves } F, f \text{ is } k \text{-times diff.} \}$ + +\remark $\S(F)$ is the Nullspace of a lin. map: $f$ to $D(f)$: +$$ +D(f) = z_1D(f_1) + z_2D(f_2) = 0 +$$ +\subtext{ $\forall z_1,z_2 \in \C,\quad f_1,f_2 \in \S$ } + +\begin{subbox}{Solutions for complex homogeneous LDEs} + \smalltext{ $F$ s.t. $a_0,\ \ldots\ ,a_{k-1}$ continuous and complex-valued } + + \begin{enumerate} + \item $\S$ is a complex vector space, $\dim(\S) = k$ + \item $\S$ is a subspace of $\{ f \sep f: I \to \C \}$ + \item $\forall x_0 \in I, (y_0,\ldots,y_{k-1}) \in \C^k$ a unique sol. exists + \end{enumerate} +\end{subbox} + +\begin{subbox}{Solutions for real homogeneous LDEs} + \smalltext{$F$ s.t. $a_0,\ \ldots\ ,a_{k-1}$ continuous and real-valued} + + \begin{enumerate} + \item $\S$ is a real vector space, $\dim(\S) = k$ + \item $\S$ is a subspace of $\{ f \sep f: I \to \R \}$ + \item $\forall x_0 \in I, (y_0,\ldots,y_{k-1}) \in \R^k$ a unique sol. exists + \end{enumerate} +\end{subbox} + +\definition \textbf{Inhomogeneous Solution Space}\\ +$\S_b(F) := \{ f + f_0 \sep f \in \S(F),\ f_0 \text{ is a particular sol.} \}$\\ +\subtext{Note: This is only a vector space if $b = 0$, where $\S_b = \S$.} + +\begin{subbox}{Solutions for real inhomogeneous LDEs} + \smalltext{$F$ s.t. $a_0,\ \ldots\ ,a_{k-1}$ continuous, $b: I \to \C$} + + \begin{enumerate} + \item $\forall x_0 \in I, (y_0,\ldots,y_{k-1}) \in \C^k$ a unique sol. exists + \item If $b, a_i$ are real-valued, a real-valued sol. exists. + \end{enumerate} +\end{subbox} + +\remark \textbf{Applications of Linearity}\\ +If $f_1$ solves $F$ for $b_1$, and $f_2$ for $b_2$: $f_1 + f_2$ solves $b_1 + b_2$. \\ +Follows from: $D(f_1) + D(f_2) = b_1 + b_2$. + +\newpage +\subsection{Finding Solutions: First Order} +\subtext{ $I \subset \R, \quad a,b: I \to \R$ } +$$ y' + ay = b $$ +Approach: +\begin{enumerate} + \item Hom. Solution: $y' + ay = 0$ using $f_1 = ke^{-A(x)}$\\ + \subtext{Note that $\S$ has $\dim(\S) = 1$, so $f_1 \neq 0$ is a Basis for $\S$} + \item Part. Solution: $f_0 \in \S_b$ using Variance of Parameters +\end{enumerate} +Solutions: $ f_0 + zf_1 \quad \text{ for } z \in \C $ + +\begin{subbox}{Explicit Solution for 1st Order LDEs} + \smalltext{$A(x)$ is a primitive of $a$, $f(x_0) = y_0$} + \begin{align*} + f(x) &= z \cdot \exp(-A(x)) \\ + f(x) &= y_0 \cdot \exp(A(x_0) - a(x)) + \end{align*} +\end{subbox} + + + diff --git a/semester3/analysis-ii-rb/util/helpers.tex b/semester3/analysis-ii-rb/util/helpers.tex new file mode 100644 index 0000000..a98c347 --- /dev/null +++ b/semester3/analysis-ii-rb/util/helpers.tex @@ -0,0 +1,68 @@ +% TC boxes +\tcbset { + base/.style={ + boxrule=0mm, + left=1.75mm, + arc=2mm, + colbacktitle=black!10!white, + coltitle=black, + fonttitle=\bfseries, + toptitle=0.75mm, + bottomtitle=0.25mm, + title={#1} + } +} +\newtcolorbox{subbox}[1]{ + colframe=black!20!white, + base={#1} +} + +% Math helpers +\def\limxo{\lim_{x\to 0}} +\def\limxi{\lim_{x\to\infty}} +\def\limxn{\lim_{x\to-\infty}} +\def\sumk{\sum_{k=1}^\infty} +\def\sumn{\sum_{n=0}^\infty} +\def\dx{\text{ d}x} + +\def\R{\mathbb{R}} +\def\Q{\mathbb{Q}} +\def\N{\mathbb{N}} +\def\C{\mathbb{C}} +\def\Z{\mathbb{Z}} + +\def\S{\mathcal{S}} + +\def\Def{\overset{\text{def.}}{\iff}} + +\def \cgeq{\succcurlyeq} +\def \cleq{\preccurlyeq} + +\def \limn{\lim\limits_{n \to \infty}} +\def \limi{\liminf\limits_{n \to \infty}} +\def \lims{\limsup\limits_{n \to \infty}} + +\def \ord{\text{ord}} +\def \sep{\ |\ } + +\def \iffdef{\overset{\text{def}}{\iff}} + +% Titles +\def \definition{\colorbox{lightgray}{Def} } +\def \notation{\colorbox{lightgray}{Notation} } +\def \remark{\colorbox{lightgray}{Remark} } +\def \theorem{\colorbox{lightgray}{Th.} } + +% For intuiton and less important notes +\def \subtext#1{ + \color{gray}\footnotesize + #1 + \color{black}\normalsize +} + +% inside tc boxes +\def \smalltext#1{ + \footnotesize + #1 + \normalsize +} diff --git a/semester3/analysis-ii-rb/util/setup.tex b/semester3/analysis-ii-rb/util/setup.tex new file mode 100644 index 0000000..c9688c2 --- /dev/null +++ b/semester3/analysis-ii-rb/util/setup.tex @@ -0,0 +1,37 @@ +\usepackage{flowfram} +\ffvadjustfalse +\setlength{\columnsep}{1cm} +\Ncolumn{3} + +% TCB boxes for important stuff +\usepackage[many]{tcolorbox} + +% Mathematical typesetting & symbols +\usepackage{amsthm, mathtools, amssymb} +\usepackage{marvosym, wasysym} +\allowdisplaybreaks + +% Tables +\usepackage{tabularx, multirow} +\usepackage{booktabs} +\renewcommand*{\arraystretch}{2} + +% Make enumerations more compact +\usepackage{enumitem} +\setitemize{itemsep=0.5pt} +\setenumerate{itemsep=0.75pt} + +% To include sketches & PDFs +\usepackage{graphicx} + +% For hyperlinks +\usepackage{hyperref} +\hypersetup{ colorlinks=true } + +% Fomatting +\usepackage{multicol} +\usepackage{parskip} % Disables new paragraph indent + +% Custom resets +\renewcommand{\arraystretch}{1.3} % Decrease row height +\renewcommand{\familydefault}{\sfdefault} \ No newline at end of file