From 8cce2e0af070345ed40d900c21bd53e80d9223a9 Mon Sep 17 00:00:00 2001 From: TobiasMeier Date: Thu, 25 Jan 2024 14:26:53 +0100 Subject: [PATCH 01/13] align env line break fixed --- src/headers/layout_formating.tex | 2 +- src/sections/Appendix.tex | 88 ++++++++++--------- src/sections/FourierAnalysis.tex | 2 + src/sections/FourierTable.tex | 7 +- src/sections/LaplaceTable.tex | 1 + src/sections/LaplaceTransformation.tex | 83 ++++++++--------- src/sections/PartialDifferentialEquations.tex | 2 + 7 files changed, 94 insertions(+), 91 deletions(-) diff --git a/src/headers/layout_formating.tex b/src/headers/layout_formating.tex index 248d301..fcf0713 100644 --- a/src/headers/layout_formating.tex +++ b/src/headers/layout_formating.tex @@ -87,5 +87,5 @@ \AtBeginDocument{% \setlength\abovedisplayskip{\eqSkipSize}% \setlength\belowdisplayskip{\eqSkipSize}% - \setlength\abovedisplayshortskip{\eqSkipSize-\baselineskip}% + \setlength\abovedisplayshortskip{0pt}% \setlength\belowdisplayshortskip{\eqSkipSize}} \ No newline at end of file diff --git a/src/sections/Appendix.tex b/src/sections/Appendix.tex index 513b3e8..3f287c3 100644 --- a/src/sections/Appendix.tex +++ b/src/sections/Appendix.tex @@ -30,6 +30,7 @@ \subsubsection{Identities} \end{align*} \subsubsection{Goniometry} +\noindent \begin{align*} \sin(x\pm y) & =\sin(x)\cos(y)\pm\cos(x)\sin(y) \\ \cos(x\pm y) & =\cos(x)\cos(y)\mp\sin(x)\sin(y) \\ @@ -60,19 +61,25 @@ \subsubsection{Goniometry} \end{align*} \subsubsection{Transformations} + \renewcommand{\arraystretch}{1.3} + \setlength\tabcolsep{6pt} % default value: 6pt + \begin{tabularx}{\linewidth}{@{}p{0.15\linewidth}ccc@{}} & $\sin$ & $\cos$ & $\tan$ \\ \cmidrule{2-4} $\sin(\alpha)=$ & & $\sqrt{1-\cos^2(\alpha)}$ & $\frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}$ \\ $\cos(\alpha)=$ & $\sqrt{1-\sin^2(\alpha)}$ & & $\frac{1}{\sqrt{1+\tan^2(\alpha)}}$ \\ - $\tan(\alpha)=$ \newline $\alpha \neq 90^\circ$ & $\frac{\sin(\alpha)}{\sqrt{1-\sin^2(\alpha)}}$ & $\frac{\sqrt{1-\cos^2(\alpha)}}{\cos(\alpha)}$ & + $\tan(\alpha)=$ \newline $\alpha \neq 90^\circ$ & $\frac{\sin(\alpha)}{\sqrt{1-\sin^2(\alpha)}}$ & $\frac{\sqrt{1-\cos^2(\alpha)}}{\cos(\alpha)}$ & \end{tabularx} + \renewcommand{\arraystretch}{1} + \setlength\tabcolsep{6pt} % default value: 6pt \subsubsection{Euler-Relation} +\noindent \begin{align*} \sin(x) & =\frac{1}{2i}(e^{ix}-e^{-ix}) \\ \cos(x) & =\frac{1}{2}(e^{ix}+e^{-ix}) \\ @@ -82,12 +89,13 @@ \subsubsection{Euler-Relation} \tanh(x) & =\frac{e^{x}-e^{-x}}{(e^{x}+e^{-x})} \\ e^{ix} & = \cos(x) + i \cdot \sin(x) \\ e^{k\pi i} & = \begin{cases} - 1 & k = 2n \\ + 1 & k = 2n \\ -1 & k = 2n + 1 \end{cases} \quad n \in \mathbb{N} \end{align*} \subsubsection{Inverse functions} +\noindent \begin{align*} \sin(\arccos(x)) & = \sqrt{1-x^2} \\ \cos(\arcsin(x)) & =\sqrt{1-x^2} \\ @@ -131,6 +139,7 @@ \subsection{Derivatives} \end{align*} \subsubsection{Common Derivatives} +\noindent \begin{align*} & \frac{d}{dx}x^n & & =nx^{n-1} \\ & \frac{d}{dx}e^x & & =e^x \\ @@ -165,15 +174,15 @@ \subsection{Integral calculus} \textbf{Partial Integration} \begin{align*} - & \int_a^b f'\cdot g\left.dx=(f\cdot g)\right|_a^b-\int_a^b f\cdot g'\;dx \\ + & \int_a^b f'\cdot g\left.dx=(f\cdot g)\right|_a^b-\int_a^b f\cdot g'\;dx \\ & \int_a^b f\cdot g\left.dx=(F\cdot g)\right|_a^b-\int_a^b F\cdot g^{\prime}\;dx \end{align*} \textbf{Substitution} \begin{align*} - & \text{Substitution }u=g(x){:} \\ - & \int_a^b f(g(x))g^{\prime}(x)\;dx=\int_{g(a)}^{g(b)}f(u)\;du \\\\ - & \text{Substitution }x=g(y){:} \\ + & \text{Substitution }u=g(x){:} \\ + & \int_a^b f(g(x))g^{\prime}(x)\;dx=\int_{g(a)}^{g(b)}f(u)\;du \\\\ + & \text{Substitution }x=g(y){:} \\ & \int_a^b f(x)\;dx=\int_{g^{-1}(a)}^{g^{-1}(b)}f(g(y))g^{\prime}(y)\;dy \end{align*} \textbf{Common Substitutions}:\par @@ -196,6 +205,7 @@ \subsection{Integral calculus} \subsubsection{Integration of trigonometric functions} +\noindent \begin{small} \begin{align*} & \int \sin(x) dx & & = -\cos(x) + C \\ @@ -245,31 +255,32 @@ \subsubsection{Integration of trigonometric functions} & \int \operatorname{arctanh}(x)dx & & = x \cdot \operatorname{arctanh}(x) +\frac{1}{2}\ln(1-x^2)+C \end{align*} \begin{align*} - & \int \sin(ax)\cos(ax) dx & & = -\frac{\cos^2(ax)}{2a}+C \\ - & \int \sin^2(x)\cos(x)dx & & =\frac{1}{3}\sin^3(x)+C \\ - & \int \sin(x)\cos^2(x)dx & & =-\frac{1}{3}\cos^3(x)+C \\ - & \int \sin^2(x)\cos^2(x)dx & & =\frac{1}{32}(4x-\sin(4x))+C \\ - & \int \sin^n(ax)\cdot \cos(ax)dx & & =\frac{\sin^{n+1}(ax)}{(n+1)a}+C \\ - & \int \sin(ax)\cdot \cos^n(ax)dx & & =-\frac{\cos^{n+1}(ax)}{(n+1)a}+C \\ + & \int \sin(ax)\cos(ax) dx & & = -\frac{\cos^2(ax)}{2a}+C \\ + & \int \sin^2(x)\cos(x)dx & & =\frac{1}{3}\sin^3(x)+C \\ + & \int \sin(x)\cos^2(x)dx & & =-\frac{1}{3}\cos^3(x)+C \\ + & \int \sin^2(x)\cos^2(x)dx & & =\frac{1}{32}(4x-\sin(4x))+C \\ + & \int \sin^n(ax)\cdot \cos(ax)dx & & =\frac{\sin^{n+1}(ax)}{(n+1)a}+C \\ + & \int \sin(ax)\cdot \cos^n(ax)dx & & =-\frac{\cos^{n+1}(ax)}{(n+1)a}+C \\ & \int \frac{\cos(ax)}{\sin^n(ax)}dx & & = -\frac{1}{(n-1)a \cdot \sin^{n-1}(ax)}+C \end{align*} \end{small} \subsubsection{Integration of trig.\ functions on special intervals} +\noindent \begin{align*} \int_{-L}^L \cos\left(\frac{n\pi x}{L}\right)\cdot \cos\left(\frac{m\pi x}{L}\right) dx & =\begin{cases} 0 & \text{if } n\neq m \\ L & \text{if } n=m \\ 2L & \text{if } n=m=0 - \end{cases} \\ + \end{cases} \\ \int_{-L}^L \sin\left(\frac{n\pi x}{L}\right)\cdot \sin\left(\frac{m\pi x}{L}\right) dx & =\begin{cases} - 0 & \text{if }n\neq m \\ + 0 & \text{if }n\neq m \\ L & \text{if } n=m\neq 0 \end{cases} \\ \int_{-L}^L \sin\left(\frac{n\pi x}{L}\right)\cdot \cos\left(\frac{m\pi x}{L}\right) dx & =0\;\;\forall\; n,m \end{align*} \begin{align*} - \sin(nx)\Big|_0^{2\pi} & =0 & & \cos(nx)\Big|_0^{2\pi}=0 \\ + \sin(nx)\Big|_0^{2\pi} & =0 & & \cos(nx)\Big|_0^{2\pi}=0 \\ x\sin(nx)\Big|_0^{2\pi} & =0 & & x\cos(nx)\Big|_0^{2\pi}=2\pi\neq0 \end{align*} \begin{align*} @@ -282,9 +293,9 @@ \subsubsection{Integration of trig.\ functions on special intervals} \renewcommand{\arraystretch}{1.5} \setlength\tabcolsep{8pt} % default value: 6pt \begin{tabularx}{\linewidth}{@{}lccccccc@{}} - & $\int\limits_0^{\frac{\pi}{4}} $ & $\int\limits_0^{\frac{\pi}{2}}$ & $\int\limits_0^{\pi}$ & $\int\limits_0^{2\pi}$ & $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} $ & $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} $ & $\int\limits_{-\pi}^{\pi}$ \\ + & $\int\limits_0^{\frac{\pi}{4}} $ & $\int\limits_0^{\frac{\pi}{2}}$ & $\int\limits_0^{\pi}$ & $\int\limits_0^{2\pi}$ & $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} $ & $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} $ & $\int\limits_{-\pi}^{\pi}$ \\ \cmidrule{2-8} - $\sin$ & $\frac{\sqrt{2}-1}{\sqrt{2}}$ & 1 & 2 & 0 & 0 & 0 & 0 \\ + $\sin$ & $\frac{\sqrt{2}-1}{\sqrt{2}}$ & 1 & 2 & 0 & 0 & 0 & 0 \\ $\sin^2$ & $\frac{\pi-2}{8}$ & $\frac{\pi}{4}$ & $\frac{\pi}{2}$ & $\pi$ & $\frac{\pi-2}{4}$ & $\frac{\pi}{2}$ & $\pi$ \\ $\sin^3$ & $\frac{8-5\sqrt{2}}{12}$ & $\frac{2}{3}$ & $\frac{4}{3}$ & 0 & 0 & 0 & 0 \\ $\sin^4$ & $\frac{3\pi-8}{32}$ & $\frac{3\pi}{16}$ & $\frac{3\pi}{8}$ & $\frac{3\pi}{4}$ & $\frac{3\pi-8}{16}$ & $\frac{3\pi}{8}$ & $\frac{3\pi}{4}$ \\ @@ -295,12 +306,12 @@ \subsubsection{Integration of trig.\ functions on special intervals} $\sin \cdot \cos$ & $\frac{1}{4}$ & $\frac{1}{2}$ & 0 & 0 & 0 & 0 & 0 \\ $\sin^2 \cdot \cos$ & $\frac{1}{6\sqrt{2}}$ & $\frac{1}{3}$ & 0 & 0 & $\frac{1}{3\sqrt{2}}$ & $\frac{2}{3}$ & 0 \\ $\sin \cdot \cos^2$ & $\frac{4-\sqrt{2}}{12}$ & $\frac{1}{3}$ & $\frac{2}{3}$ & 0 & 0 & 0 & 0 \\ - \end{tabularx} \renewcommand{\arraystretch}{1} \setlength\tabcolsep{6pt} % default value: 6pt \subsubsection{Intergation of power, rational, exponential and logarithmic functions} +\noindent \begin{footnotesize} \begin{align*} & \int {(ax+b)}^n dx & & =\frac{{(ax+b)}^{n+1}}{(n+1)a}+C \\ @@ -358,6 +369,7 @@ \subsubsection{Intergation of power, rational, exponential and logarithmic funct \end{footnotesize} \subsubsection{Integration over special intervals} +\noindent \begin{align*} & \int_{-\infty}^{\infty}\frac{1}{1+x^2}dx & & = \pi \\ & \int_0^{\infty} e^{-ax}x^n dx & & =\frac{n!}{a^{n+1}},\quad a>0 \\ @@ -369,6 +381,7 @@ \subsubsection{Integration over special intervals} \subsection{Various functions and their properties} \subsubsection{even and odd functions} +\noindent \begin{gather*} even \cdot even\;\widehat{=}\; odd \cdot odd \;\widehat{=}\; even \\ even \cdot odd \;\widehat{=}\; odd \\ @@ -380,6 +393,7 @@ \subsubsection{even and odd functions} \end{gather*} \subsubsection{Logarithms} +\noindent \begin{align*} \ln(uv) & =\ln(u)+\ln(v) \\ \ln\left(\frac{u}{v}\right) & =\ln(u)-\ln(v) \\ @@ -401,6 +415,7 @@ \subsubsection{Gamma functions} \end{align*} \subsubsection{Body properties} +\noindent \begin{align*} \text{Sphere volume V } & = \frac{4}{3}\pi r^3 \\ \text{Sphere surface A } & = 4\pi r^2 \\ @@ -417,19 +432,6 @@ \subsubsection{Complex numbers} & $\varphi=\arccos\left(\frac{x}{r}\right)=\arcsin\left(\frac{y}{r}\right)=\tan\left(\frac{y}{x}\right)$ \\ Exp.\ form: & $z=r\; e^{i\varphi}$ \\ & $e^{i\; \varphi}=\cos(\varphi)+i\; \sin(\varphi)$ - - - - - - - - - - - - - \end{tabular} \textbf{Operations} @@ -461,22 +463,22 @@ \subsection{Dirichlet Superposition}\label{diri_superpos} \end{equation*} \textbf{Solution for A} \begin{align*} - u_1(x,y) & =\sum_{n=1}^\infty A_n\sin\left(\frac{n\pi x}a\right)\sinh\left(\frac{n\pi(b-y)}a\right) \\ + u_1(x,y) & =\sum_{n=1}^\infty A_n\sin\left(\frac{n\pi x}a\right)\sinh\left(\frac{n\pi(b-y)}a\right) \\ A_n & =\frac2{a\sinh\left(\frac{n\pi b}a\right)}\int_0^{a}f_1(x)\sin\left(\frac{n\pi x}a\right)\mathrm{d}x \end{align*} \textbf{Solution for B} \begin{align*} - u_2(x,y) & =\sum_{n=1}^\infty B_n\sin\left(\frac{n\pi x}a\right)\sinh\left(\frac{n\pi y}a\right) \\ + u_2(x,y) & =\sum_{n=1}^\infty B_n\sin\left(\frac{n\pi x}a\right)\sinh\left(\frac{n\pi y}a\right) \\ B_n & =\frac2{a\sinh\left(\frac{n\pi b}a\right)}\int_0^{a}f_2(x)\sin\left(\frac{n\pi x}a\right)\mathrm{d}x \end{align*} \textbf{Solution for C} \begin{align*} - u_3(x,y) & =\sum_{n=1}^\infty C_n\sinh\left(\frac{n\pi(a-x)}a\right)\sin\left(\frac{n\pi y}b\right) \\ + u_3(x,y) & =\sum_{n=1}^\infty C_n\sinh\left(\frac{n\pi(a-x)}a\right)\sin\left(\frac{n\pi y}b\right) \\ C_n & =\frac2{b\sinh\left(\frac{n\pi a}b\right)}\int_0^{b}g_1(y)\sin\left(\frac{n\pi y}b\right)\mathrm{d}y \end{align*} \textbf{Solution for D} \begin{align*} - u_4(x,y) & =\sum_{n=1}^\infty D_n\sinh\left(\frac{n\pi x}b\right)\sin\left(\frac{n\pi y}b\right) \\ + u_4(x,y) & =\sum_{n=1}^\infty D_n\sinh\left(\frac{n\pi x}b\right)\sin\left(\frac{n\pi y}b\right) \\ D_n & =\frac2{b\sinh\left(\frac{n\pi a}b\right)}\int_0^{b}g_2(y)\sin\left(\frac{n\pi y}b\right)\mathrm{d}y \end{align*} @@ -522,15 +524,15 @@ \subsubsection{Inhomogeneous Linear 1st Order ODE} \end{align*} and \begin{align*} - u_p(x) & = S(x) \cdot e^{A(x)} \\ + u_p(x) & = S(x) \cdot e^{A(x)} \\ S(x) & = \int s(x)\cdot e^{-A(x)} \,dx \end{align*} \textbf{Solutions to some common linear 1st order ODEs} \begin{align*} - u'(x) & = a \cdot u(x) & \Rightarrow & \quad u(x)=Ce^{ax} & & \\ - u'(x) & = a \cdot u(x) + b & \Rightarrow & \quad u(x)=Ce^{ax}-\frac{b}{a} & & \\ - u'(x) & = f(x) \cdot u(x) & \Rightarrow & \quad u(x) = Ce^{F(x)} & & + u'(x) & = a \cdot u(x) & \Rightarrow & \quad u(x)=Ce^{ax} & & \\ + u'(x) & = a \cdot u(x) + b & \Rightarrow & \quad u(x)=Ce^{ax}-\frac{b}{a} & & \\ + u'(x) & = f(x) \cdot u(x) & \Rightarrow & \quad u(x) = Ce^{F(x)} & & \end{align*} with \begin{equation*} @@ -604,11 +606,11 @@ \subsubsection{Homogeneous Linear n-th Order ODE with Constant Coefficients}\lab u(x) = C_1e^x + C_2xe^x + C_3x^2e^x + C_4e^{3x} \end{equation*} \end{enumerate} - + \hrule - + 4-th order ODE with complex roots - + \begin{equation*} u^{(4)} + 3u'' - 4u = 0 \end{equation*} @@ -650,7 +652,7 @@ \subsubsection{Inhomogeneous linear n-th order ODE with constant coefficients} \end{equation*} \item \textbf{Polynomial} \begin{align*} - s(x) & = p_m(x) \\ + s(x) & = p_m(x) \\ u_p(x) & = q_m(x) = Ax^m + Bx^{m-1} + \ldots \end{align*} Special case: diff --git a/src/sections/FourierAnalysis.tex b/src/sections/FourierAnalysis.tex index 12427b2..fa285c9 100644 --- a/src/sections/FourierAnalysis.tex +++ b/src/sections/FourierAnalysis.tex @@ -29,6 +29,7 @@ \subsubsection{Properties of Periodic Functions} \end{enumerate} \begin{examplesection}[Example of periodic functions] + \noindent \begin{gather*} f(x) = \cos(3x) \qquad g(x) = \cos(4x) \\ h(x) = f(x) + g(x) \\\\ @@ -47,6 +48,7 @@ \subsection{Trigonometric System} is called the $trigonometric$ $system$. The functions in the trigonometric system are $2L$ periodic. A series constructed from these functions is $2L$ periodic too if it converges. \subsubsection{Orthogonality Properties} +\noindent \begin{align*} \int_{-L}^L\cos\left(\frac{n\pi}Lx\right)\cos\left(\frac{m\pi}Lx\right)dx&=\begin{cases}0&n\neq m\\L&n=m\neq0\\2L&n=m=0.\end{cases} \\ \int_{-L}^L\sin\left(\frac{n\pi}Lx\right)\sin\left(\frac{m\pi}Lx\right)dx&=\begin{cases}0&n\neq m\\L&n=m\neq0.\end{cases} \\ diff --git a/src/sections/FourierTable.tex b/src/sections/FourierTable.tex index 622a745..54e839b 100644 --- a/src/sections/FourierTable.tex +++ b/src/sections/FourierTable.tex @@ -1,16 +1,17 @@ \subsection{Fourier Transform Table} \subsubsection{Known Transforms}\label{known_transforms} +\noindent \begin{align*} e^{-ax^2} & \;\laplace\; \frac1{\sqrt{2a}}e^{\frac{-w^2}{4a}} \\ - xe^{-ax^2} & \;\laplace\; -\frac{i\omega}{{(2a)}^{3/2}}e^{-\frac{\omega^2}{4a}} \\ + xe^{-ax^2} & \;\laplace\; -\frac{i\omega}{{(2a)}^{3/2}}e^{-\frac{\omega^2}{4a}} \\ x^2e^{-ax^2} & \;\laplace\; \left( 1-\frac{\omega^2}{2a} \right) \frac{1}{\sqrt{8a^3}}e^{-\frac{\omega^2}{4a}} \\ \frac x{{(2b)}^{3/2}}e^{-\frac{x^2}{4b}} & \;\laplace\; i\omega e^{-b\omega^2} \\ \int_{\mathbb{R}}g(x)dx & = \sqrt{2\pi}\widehat{g}(0) \end{align*} \subsubsection{Fourier transform of time continuous signals} -{\footnotesize \textbf{Remark} The following fourier transformations are taken from \textit{Formelsammlung Signal- und Systemtheorie I} by Prof. Dr. Helmut Bölskei, -where the Fourier Transform is defined without the factor $\frac{1}{\sqrt{2\pi}}$.} +{\footnotesize \textbf{Remark} The following fourier transformations are taken from \textit{Formelsammlung Signal- und Systemtheorie I} by Prof. Dr. Helmut Bölskei, + where the Fourier Transform is defined without the factor $\frac{1}{\sqrt{2\pi}}$.} \begin{equation*} \widehat{x}(f) =\int_{-\infty}^{\infty} x(t) e^{-2\pi ift}\; dt \end{equation*} diff --git a/src/sections/LaplaceTable.tex b/src/sections/LaplaceTable.tex index c6bec93..606033b 100644 --- a/src/sections/LaplaceTable.tex +++ b/src/sections/LaplaceTable.tex @@ -1,4 +1,5 @@ \subsection{Laplace Transform Table} +\noindent \begin{align*} f(t) \; & \laplace\; F(s) \\ 1 \; & \laplace\; \frac{1}{s} \\ diff --git a/src/sections/LaplaceTransformation.tex b/src/sections/LaplaceTransformation.tex index 278d913..5fa064c 100644 --- a/src/sections/LaplaceTransformation.tex +++ b/src/sections/LaplaceTransformation.tex @@ -46,15 +46,14 @@ \subsection{Linearity} \end{equation*} \hrule{} Let $F(s)=\frac{a}{bs+c}; a,b,c \in \mathbb{R}$, then - \begin{equation*} - F(s)=\frac{a}{bs+c}=\frac{\frac{a}{b}}{s+c/b}=\frac{\frac{a}{b}}{s-(-\frac{c}{b})} - \end{equation*} - \begin{equation*} + \begin{gather*} + F(s)=\frac{a}{bs+c}=\frac{\frac{a}{b}}{s+c/b}=\frac{\frac{a}{b}}{s-(-\frac{c}{b})} \\ f(t)=\mathcal{L}^{-1}[F(s)]=\frac{a}{b}\mathcal{L}^{-1}\left[\frac{1}{s-(-\frac{c}{b})}\right]=\frac{a}{b}e^{-\frac{c}{b}t} - \end{equation*} + \end{gather*} \end{examplesection} \subsection{s-shifting} +\noindent \begin{equation*} e^{at}\cdot f(t)\:\laplace\:F(s-a) \end{equation*} @@ -76,6 +75,27 @@ \subsection{Differentiation}\label{ssec:laplaceDiff} \mathcal{L}[f''](s) & =s^2\mathcal{L}[f](s)-sf(0)-f'(0) \\ \mathcal{L}[f'''](s) & =s^3\mathcal{L}[f](s)-s^2f(0)-sf'(0)-f''(0) % chktex 23 \end{align*} + +\subsection{Solving initial value problems} +\begin{enumerate} + \item Transform the ODE to the s-domain ($\mathcal{L}[y]=Y$)\\ + $\Rightarrow$ Insert initial conditions\\ + $\quad\qquad y''+ay'+by=r(t)$\\ + $\quad\qquad(s^2Y-sy(0)-y'(0))+a(sY-y(0))+bY=R(s)$ + \item Solve the equation for $Y$ \\ + $\quad\qquad (s^2+as+b)Y=R(s)+sy(0)+y'(0)+ay(0)$ + \item Transform the equation back to the time-domain $y=\mathcal{L}^{-1}[Y]$ +\end{enumerate} +Special case:\\ +If the initial values are given like this $y(a),y'(a),\dots$: +\begin{enumerate}[label=\alph*] + \item Substitute: $t=\tilde{t}+a$ + \item $y''+ay'+by=r(t) \Rightarrow \tilde{y}''+a\tilde{y}'+b\tilde{y}=r(\tilde{t}+a)$\\ + $\tilde{y}(0)=y(a),\:\tilde{y}'(0)=y'(a),\dots$ + \item Solve the ODE as described above $\Rightarrow \tilde{Y} \rightarrow \tilde{y}(\tilde{t})$ + \item Resubstitute: $\tilde{t}=t-a$; $\tilde{y}(\tilde{t})\rightarrow y(t)$ +\end{enumerate} + \begin{examplesection}[Example] Give the ODE $y''-y'+y=0,\;y(0)=0,\;y'(0)=1$ \begin{align*} @@ -89,23 +109,21 @@ \subsection{Differentiation}\label{ssec:laplaceDiff} \subsection{Differentiation of transforms} Let $f$ be a piecewise continuous function that satisfies the growth condition. Then -\begin{equation*} - \mathcal{L}'[f(t)] = \frac{d}{ds}\mathcal{L}[f(t)] = -\mathcal{L}[t f(t)] -\end{equation*} -\begin{equation*} +\begin{gather*} + \mathcal{L}'[f(t)] = \frac{d}{ds}\mathcal{L}[f(t)] = -\mathcal{L}[t f(t)] \\ \mathcal{L}^{-1}[F'(s)] = -tf(t) -\end{equation*} +\end{gather*} \subsection{Heaviside function} \begin{tabular}{ m{5cm} m{3.4cm} } \begin{equation*} u(t):= \begin{cases} - 1 & \text{if}\;t>0 \\ + 1 & \text{if}\;t>0 \\ 0 & \text{if}\;t<0 \end{cases} \end{equation*} - & + & \begin{tikzpicture} \begin{axis}[ width=\linewidth, @@ -139,11 +157,11 @@ \subsection{Heaviside function} \begin{equation*} u(t-a):= \begin{cases} - 1 & \text{if}\;t>a \\ + 1 & \text{if}\;t>a \\ 0 & \text{if}\;t Date: Fri, 16 Feb 2024 18:09:17 +0100 Subject: [PATCH 02/13] template repo as submodule added --- .gitmodules | 3 +++ src/template | 1 + 2 files changed, 4 insertions(+) create mode 100644 .gitmodules create mode 160000 src/template diff --git a/.gitmodules b/.gitmodules new file mode 100644 index 0000000..6fa2537 --- /dev/null +++ b/.gitmodules @@ -0,0 +1,3 @@ +[submodule "src/template"] + path = src/template + url = https://github.com/MeierTobias/eth-summary-template.git diff --git a/src/template b/src/template new file mode 160000 index 0000000..d617a9b --- /dev/null +++ b/src/template @@ -0,0 +1 @@ +Subproject commit d617a9be0578667f3f76f8567b2b7186dbd9048b From 4226f1c6d5b979e0e9bb536684e3142cc6705ece Mon Sep 17 00:00:00 2001 From: Tobias Meier Date: Fri, 16 Feb 2024 19:49:49 +0100 Subject: [PATCH 03/13] submodule template implemented --- src/headers/code_header.tex | 80 ---------------------- src/headers/conditional_formatting.tex | 11 ---- src/headers/flags.tex | 2 - src/headers/layout_formating.tex | 91 -------------------------- src/headers/math_header.tex | 32 --------- src/headers/standard_header.tex | 75 --------------------- src/main.tex | 69 ++++++++----------- src/template | 2 +- src/var.tex | 18 +++++ 9 files changed, 47 insertions(+), 333 deletions(-) delete mode 100644 src/headers/code_header.tex delete mode 100644 src/headers/conditional_formatting.tex delete mode 100644 src/headers/flags.tex delete mode 100644 src/headers/layout_formating.tex delete mode 100644 src/headers/math_header.tex delete mode 100644 src/headers/standard_header.tex create mode 100644 src/var.tex diff --git a/src/headers/code_header.tex b/src/headers/code_header.tex deleted file mode 100644 index 1744fca..0000000 --- a/src/headers/code_header.tex +++ /dev/null @@ -1,80 +0,0 @@ -%-----------------------------------------------------% -% for colours -\usepackage{color} -% from code expert "ce_" -\definecolor{ce_yellow}{rgb}{0.902,0.859,0.455} -\definecolor{ce_gray}{rgb}{0.459,0.443,0.369} -\definecolor{ce_lime}{rgb}{0.459,0.816,0.180} -\definecolor{ce_pink}{rgb}{0.976,0.149,0.447} -\definecolor{ce_cyan}{rgb}{0.40,0.851,0.937} -\definecolor{ce_violet}{rgb}{0.545,0.506,1.00} -\definecolor{ce_back}{rgb}{0.153,0.157,0.133} -\definecolor{ce_white}{rgb}{1,1,1} -%----------------------------------------------------% -% as close to CodeExpert as i could get it -\usepackage{listings} - -\lstdefinestyle{CodeExpert}{ - language=C++, - basicstyle=\ttfamily\linespread{0.8}\color{ce_white}, - numbers=none, - aboveskip=0mm, - belowskip=0mm, - frame = none, - numberstyle=\tiny\color{ce_grey}, - backgroundcolor = \color{ce_back}, - keywordstyle=\color{ce_cyan}, - commentstyle=\color{ce_gray}, - stringstyle=\color{ce_yellow}, - morecomment=[n][\color{ce_pink}]{\#}{\ }, - literate= - *{./}{{{\color{ce_pink}./}}}2 - {.^}{{{\color{ce_pink}.\^{}}}}2 - {=}{{{\color{ce_pink}=}}}1 - {+}{{{\color{ce_pink}+}}}1 - {*}{{{\color{ce_pink}*}}}1 - {-}{{{\color{ce_pink}-}}}1 - {&}{{{\color{ce_pink}&}}}1 - {<<}{{{\color{ce_pink}<<}}}2 - {>>}{{{\color{ce_pink}>>}}}2 - {<}{{{\color{ce_pink}<}}}1 - {>}{{{\color{ce_pink}>}}}1 - {->}{{{\color{ce_pink}->}}}2 - {1}{{{\color{ce_violet}1}}}1 - {2}{{{\color{ce_violet}2}}}1 - {3}{{{\color{ce_violet}3}}}1 - {4}{{{\color{ce_violet}4}}}1 - {5}{{{\color{ce_violet}5}}}1 - {6}{{{\color{ce_violet}6}}}1 - {7}{{{\color{ce_violet}7}}}1 - {8}{{{\color{ce_violet}8}}}1 - {9}{{{\color{ce_violet}9}}}1 - {0}{{{\color{ce_violet}0}}}1 - {this}{{{\color{ce_lime}this}}}1 - {if}{{{\color{ce_pink}if}}}1 - {do}{{{\color{ce_pink}do}}}1 - {for}{{{\color{ce_pink}for}}}1 - {else}{{{\color{ce_pink}else}}}1 - {then}{{{\color{ce_pink}then}}}1 - {break}{{{\color{ce_pink}break}}}1 - {continue}{{{\color{ce_pink}continue}}}1 - {public}{{{\color{ce_pink}public}}}1 - {private}{{{\color{ce_pink}private}}}1 - {while}{{{\color{ce_pink}while}}}1 - {continue}{{{\color{ce_pink}continue}}}1 - {nullptr}{{{\color{ce_violet}nullptr}}}1 - {NULL}{{{\color{ce_violet}NULL}}}1, -} - -\lstset% - { - basicstyle=\ttfamily, - frame=tb, - aboveskip=1mm, - belowskip=1mm, - showstringspaces=true, - columns=flexible, - breaklines=true, - breakatwhitespace=true, - tabsize=2, -} \ No newline at end of file diff --git a/src/headers/conditional_formatting.tex b/src/headers/conditional_formatting.tex deleted file mode 100644 index 355bada..0000000 --- a/src/headers/conditional_formatting.tex +++ /dev/null @@ -1,11 +0,0 @@ -\if 0\udlayout{} - % A4 landscape - \def\udorientationmode{landscape} - \def\udpapersize{a4paper} - \def\udfontsize{10pt} -\else - % A3 portrait - \def\udorientationmode{portrait} - \def\udpapersize{a3paper} - \def\udfontsize{10pt} -\fi \ No newline at end of file diff --git a/src/headers/flags.tex b/src/headers/flags.tex deleted file mode 100644 index 89429c3..0000000 --- a/src/headers/flags.tex +++ /dev/null @@ -1,2 +0,0 @@ -\def\udlayout{1} % 0 = 4A landscape, 1 = A3 portrait -\def\udexamples{1} % 0 = don't show examples, 1 = show examples diff --git a/src/headers/layout_formating.tex b/src/headers/layout_formating.tex deleted file mode 100644 index fcf0713..0000000 --- a/src/headers/layout_formating.tex +++ /dev/null @@ -1,91 +0,0 @@ -% colors -\usepackage{color,xcolor} -\definecolor{sectionColor}{HTML}{1c59c9} -\definecolor{subSectionColor}{HTML}{568ae8} -\definecolor{subSubsectionColor}{HTML}{87a9e8} -\definecolor{titleTextColor}{RGB}{255,255,255} - -% set the size of a section -\usepackage{parskip} -\setlength{\parindent}{0pt} -\setlength{\parskip}{0pt} - -% used for colored boxes -\usepackage[many]{tcolorbox} - -% customize the headers -\usepackage[explicit]{titlesec} -\usepackage{titletoc} - -\titleformat{\section} -{\normalfont\bfseries\fontfamily{lmss}\selectfont} -{} -{0pt} -{\begin{tcolorbox}[ - enhanced, - boxrule=0pt, - arc=0pt, - outer arc=0pt, - left=0pt, - right=0pt, - top=0pt, - bottom=0pt, - nobeforeafter, - interior code={\fill[overlay,sectionColor] (frame.north west) rectangle (frame.south east);}, - ]\textcolor{titleTextColor}{\thesection\hspace{0.5em}#1} - \end{tcolorbox}}[{\startcontents[section]}] -\titlespacing{\section}{0pt}{1pt}{0pt} - -\titleformat{\subsection} -{\normalfont\bfseries\fontfamily{lmss}\selectfont} -{} -{0pt} -{\begin{tcolorbox}[ - enhanced, - boxrule=0pt, - arc=0pt, - outer arc=0pt, - left=0pt, - right=0pt, - top=0pt, - bottom=0pt, - nobeforeafter, - interior code={\fill[overlay,subSectionColor] (frame.north west) rectangle (frame.south east);}, - ]\textcolor{titleTextColor}{\thesubsection\hspace{0.5em}#1} - \end{tcolorbox}} -\titlespacing{\subsection}{0pt}{1pt}{0pt} - -\titleformat{\subsubsection} -{\normalfont\bfseries\fontfamily{lmss}\selectfont} -{} -{0pt} -{\begin{tcolorbox}[ - enhanced, - boxrule=0pt, - arc=0pt, - outer arc=0pt, - left=0pt, - right=0pt, - top=0pt, - bottom=0pt, - nobeforeafter, - interior code={\fill[overlay,subSubsectionColor] (frame.north west) rectangle (frame.south east);}, - ]\textcolor{titleTextColor}{\thesubsubsection\hspace{0.5em}#1} - \end{tcolorbox}} -\titlespacing{\subsubsection}{0pt}{1pt}{0pt} - -% Create a TOC for the current section -\newcommand{\createsectiontoc}{ - \printcontents[section]{p}{2}{}{}% -} -% Removes the page numbering from the sub- and subsub-section -\titlecontents{subsection}[0.2em]{}{\thecontentslabel\hspace{1em}}{}{}[] -\titlecontents{subsubsection}[3em]{}{\thecontentslabel\hspace{1em}}{}{}[] - -% set the spacing above and below the equations -\newlength{\eqSkipSize} \setlength{\eqSkipSize}{6pt} -\AtBeginDocument{% - \setlength\abovedisplayskip{\eqSkipSize}% - \setlength\belowdisplayskip{\eqSkipSize}% - \setlength\abovedisplayshortskip{0pt}% - \setlength\belowdisplayshortskip{\eqSkipSize}} \ No newline at end of file diff --git a/src/headers/math_header.tex b/src/headers/math_header.tex deleted file mode 100644 index 24a82a7..0000000 --- a/src/headers/math_header.tex +++ /dev/null @@ -1,32 +0,0 @@ -%Mathematik-Pakete -\usepackage{amsmath, amstext, amssymb, mathtools, esint, polynom, trfsigns, pgfplots} -%Spec. version of pgfplots (rel. new version 2023) -\pgfplotsset{compat=1.18} -\usetikzlibrary{backgrounds} -\usepackage{bm} -%Seitenumbruch in align-Umgebung erlauben -\allowdisplaybreaks{} -% - -%Definition der Umgebung "example" -\newenvironment{example} -{\begin{itshape} \begin{small}} - {\end{small} \end{itshape}} -% -%Definition der Umgebung "annotation" -\newenvironment{annotation}[1] -{\begin{itshape} \begin{small} \textbf{#1} \begin{itemize}} - {\end{itemize} \end{small} \end{itshape}} -% -%Definition der Umgebung "eq" -\newenvironment{eq} -{\begin{equation*}} - {\end{equation*}} -% -% Don't know what this does -\providecommand{\diff}{\mathop{} \! \mathrm{d}} -\DeclareMathOperator{\rot}{rot} -\DeclareMathOperator{\divg}{div} - -% define colors -\definecolor{mathGreen}{RGB}{0, 120, 40} \ No newline at end of file diff --git a/src/headers/standard_header.tex b/src/headers/standard_header.tex deleted file mode 100644 index 22fd7a0..0000000 --- a/src/headers/standard_header.tex +++ /dev/null @@ -1,75 +0,0 @@ -% document setup -\usepackage[left=7mm,right=7mm,top=20mm,bottom=7mm,\udorientationmode]{geometry} % margin = .. total={280mm,190 mm} % in geometry for defined size/ratio -\usepackage{multicol,multirow} -\usepackage[utf8]{inputenc} % not strictly necessary, but sets utf8 - -% enable colors -\usepackage{xcolor,color} % standard colors (blue, red, etc.https://www.namsu.de/Extra/pakete/Xcolor.html ) - -% multicol settings -\setlength{\premulticols}{1pt} -\setlength{\postmulticols}{1pt} -\setlength{\multicolsep}{1pt} -\setlength{\columnsep}{5pt} -\setlength{\columnseprule}{1pt} -\def\columnseprulecolor{\color{black}} - -% language -\usepackage[english]{babel} %choose your language - -% for images -\usepackage{graphicx} -\graphicspath{ {./images/} } - -% Images combined with texts -\usepackage{wrapfig} - -% some AsmTeX options -\usepackage{amscd, amsmath,amssymb} - -% more fine control for lists -\usepackage{enumitem} -\setlist{noitemsep} -\setlist[itemize]{leftmargin = 8pt, label = ·} -\setlist[enumerate]{leftmargin = 14pt} -%\setlist{nosep} - -% multiple line comments and testing text -\usepackage{comment} % \begin{comment} \end{comment} -\usepackage{blindtext} % inserts lorem ipsum like text - -% table that supports equations -\usepackage{tabularx} -\usepackage{booktabs} - -% Example section environment -\newenvironment{examplesection}[1][] -{ - \subsubsection{#1} - \color{blue} -} -{ -} - -\if 0\udexamples - \excludecomment{examplesection} -\fi - -% define header style -\usepackage[headsepline]{scrlayer-scrpage} -\pagestyle{scrheadings} -\ihead{Analysis III - \today} -\chead{\pagemark} -\ohead{\url{https://github.com/MeierTobias/eth-analysis-3}} -\setlength{\headsep}{5pt} - -% hyperlinks have to be included last -\usepackage{hyperref} -\hypersetup{ - colorlinks, - citecolor=blue, - filecolor=black, - linkcolor=black, - urlcolor=black, - pdftitle={Analysis III} -} \ No newline at end of file diff --git a/src/main.tex b/src/main.tex index b0ea70d..a189d76 100644 --- a/src/main.tex +++ b/src/main.tex @@ -1,41 +1,28 @@ -\input{headers/flags} -\input{headers/conditional_formatting.tex} - -\documentclass[\udpapersize,\udfontsize]{extarticle} - -\input{headers/standard_header} -\input{headers/layout_formating} -\input{headers/code_header} -\input{headers/math_header} - -\begin{document} -\begin{multicols*}{3} - - \begin{center} - \huge{Analysis III \par} - \large{HS23 ETHZ\par} - - \normalsize{Juri Pfammatter, Daniel Schweizer, Tobias Meier, Benno Kaeslin, Linard Furck, Sandro Christen, Markus Fuchs, Andre Jauch} - \end{center} - This PDF, the source code as well as the disclaimer can be found in the GitHub repository \url{https://github.com/MeierTobias/eth-analysis-3}. - - - \input{sections/LaplaceTransformation.tex} - \newpage - \input{sections/LaplaceTable.tex} - \input{sections/FourierTable.tex} - \newpage - \input{sections/FourierAnalysis.tex} - \newpage - \input{sections/PartialDifferentialEquations.tex} - \newpage - \input{sections/Appendix.tex} - - % a model columnbreak - % \vfill\null % prevents the columns content from using the full height - % \columnbreak - % - -\end{multicols*} - -\end{document} % chktex 17 \ No newline at end of file +% Define all the needed variable like it is shown in the var.tex.example +% (Just copy the var.tex.example file and remove the ".example" from the filename.) +\input{var.tex} + +\documentclass[\papersize,\fontheight]{extarticle} + +% load the template headers +\input{\templatePath/summary_headers.tex} % ChkTex 27 + +\begin{document} +\begin{multicols*}{3} + + % create the title section + \input{\templatePath/summary_title.tex} % ChkTex 27 + + \input{sections/LaplaceTransformation.tex} + \newpage + \input{sections/LaplaceTable.tex} + \input{sections/FourierTable.tex} + \newpage + \input{sections/FourierAnalysis.tex} + \newpage + \input{sections/PartialDifferentialEquations.tex} + \newpage + \input{sections/Appendix.tex} + +\end{multicols*} +\end{document} \ No newline at end of file diff --git a/src/template b/src/template index d617a9b..77213d5 160000 --- a/src/template +++ b/src/template @@ -1 +1 @@ -Subproject commit d617a9be0578667f3f76f8567b2b7186dbd9048b +Subproject commit 77213d5a617f15e7d409ec4e4df4dabe05d81054 diff --git a/src/var.tex b/src/var.tex new file mode 100644 index 0000000..90f06ad --- /dev/null +++ b/src/var.tex @@ -0,0 +1,18 @@ +% specify the relative path from the main file to the folder that contains the template repo +\newcommand{\templatePath}{template} + +% define some information about this summary +\newcommand{\summaryTitle}{Analysis III} +\newcommand{\summarySubTitle}{HS23 ETHZ} +\newcommand{\summaryAuthor}{Juri Pfammatter, Daniel Schweizer, Tobias Meier, Benno Kaeslin, Linard Furck, Sandro Christen, Markus Fuchs, Andre Jauch} +\newcommand{\repoURL}{https://github.com/MeierTobias/eth-analysis-3} +\newcommand{\summaryInfo}{This PDF, the source code as well as the disclaimer can be found in the GitHub repository \url{\repoURL}.} +\newcommand{\imagePath}{images} + +% layout options +\newcommand{\orientationmode}{portrait} % landscape or portrait +\newcommand{\papersize}{a3paper} % a4paper or a3paper +\newcommand{\fontheight}{10pt} + +% additional conditions +\newcommand{\includeexamples}{1} % 0 = don't show examples, 1 = show examples \ No newline at end of file From 1f1008d29597dcbee989d6b631ea72b14a874e79 Mon Sep 17 00:00:00 2001 From: Tobias Meier Date: Fri, 16 Feb 2024 20:07:16 +0100 Subject: [PATCH 04/13] template updated --- src/main.tex | 4 ++-- src/template | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/src/main.tex b/src/main.tex index a189d76..67c8b9e 100644 --- a/src/main.tex +++ b/src/main.tex @@ -2,7 +2,7 @@ % (Just copy the var.tex.example file and remove the ".example" from the filename.) \input{var.tex} -\documentclass[\papersize,\fontheight]{extarticle} +\documentclass[\fontheight]{extarticle} % load the template headers \input{\templatePath/summary_headers.tex} % ChkTex 27 @@ -25,4 +25,4 @@ \input{sections/Appendix.tex} \end{multicols*} -\end{document} \ No newline at end of file +\end{document} % ChkTeX 17 \ No newline at end of file diff --git a/src/template b/src/template index 77213d5..ccbb214 160000 --- a/src/template +++ b/src/template @@ -1 +1 @@ -Subproject commit 77213d5a617f15e7d409ec4e4df4dabe05d81054 +Subproject commit ccbb214c504e71c565ecdd3def13e3bbe6959242 From 89aee29a344adbdf4b582be3bf9d687e14b02bcc Mon Sep 17 00:00:00 2001 From: Tobias Meier Date: Fri, 16 Feb 2024 23:16:09 +0100 Subject: [PATCH 05/13] warnings fixed --- src/main.tex | 5 ++--- src/sections/Appendix.tex | 11 ++++++++-- src/sections/FourierAnalysis.tex | 22 +++++++++++-------- src/sections/FourierTable.tex | 2 +- src/sections/LaplaceTransformation.tex | 6 ++--- src/sections/PartialDifferentialEquations.tex | 18 +++++++-------- 6 files changed, 37 insertions(+), 27 deletions(-) diff --git a/src/main.tex b/src/main.tex index 67c8b9e..3350577 100644 --- a/src/main.tex +++ b/src/main.tex @@ -1,5 +1,3 @@ -% Define all the needed variable like it is shown in the var.tex.example -% (Just copy the var.tex.example file and remove the ".example" from the filename.) \input{var.tex} \documentclass[\fontheight]{extarticle} @@ -23,6 +21,7 @@ \input{sections/PartialDifferentialEquations.tex} \newpage \input{sections/Appendix.tex} - + \end{multicols*} + \end{document} % ChkTeX 17 \ No newline at end of file diff --git a/src/sections/Appendix.tex b/src/sections/Appendix.tex index 3f287c3..ed694e5 100644 --- a/src/sections/Appendix.tex +++ b/src/sections/Appendix.tex @@ -185,8 +185,11 @@ \subsection{Integral calculus} & \text{Substitution }x=g(y){:} \\ & \int_a^b f(x)\;dx=\int_{g^{-1}(a)}^{g^{-1}(b)}f(g(y))g^{\prime}(y)\;dy \end{align*} + \textbf{Common Substitutions}:\par + \renewcommand{\arraystretch}{1.3} + \begin{tabularx}{\linewidth}{@{}lll@{}} Integral & Subst. & Comment \\ \cmidrule{1-3} @@ -200,8 +203,8 @@ \subsection{Integral calculus} & & $dx=\frac{2dt}{1+t^2}$ \\ $f(e^x,\sinh,\cosh)$ & $t=e^x$ & $\sinh(x)=\frac{t^2-1}{2t}$ \\ \end{tabularx} -\renewcommand{\arraystretch}{1} +\renewcommand{\arraystretch}{1} \subsubsection{Integration of trigonometric functions} @@ -291,7 +294,9 @@ \subsubsection{Integration of trig.\ functions on special intervals} \renewcommand{\arraystretch}{1.5} + \setlength\tabcolsep{8pt} % default value: 6pt + \begin{tabularx}{\linewidth}{@{}lccccccc@{}} & $\int\limits_0^{\frac{\pi}{4}} $ & $\int\limits_0^{\frac{\pi}{2}}$ & $\int\limits_0^{\pi}$ & $\int\limits_0^{2\pi}$ & $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} $ & $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} $ & $\int\limits_{-\pi}^{\pi}$ \\ \cmidrule{2-8} @@ -305,9 +310,11 @@ \subsubsection{Integration of trig.\ functions on special intervals} $\cos^4$ & $\frac{8+3\pi}{32}$ & $\frac{3\pi}{16}$ & $\frac{3\pi}{8}$ & $\frac{3\pi}{4}$ & $\frac{8+3\pi}{16}$ & $\frac{3\pi}{8}$ & $\frac{3\pi}{4}$ \\ $\sin \cdot \cos$ & $\frac{1}{4}$ & $\frac{1}{2}$ & 0 & 0 & 0 & 0 & 0 \\ $\sin^2 \cdot \cos$ & $\frac{1}{6\sqrt{2}}$ & $\frac{1}{3}$ & 0 & 0 & $\frac{1}{3\sqrt{2}}$ & $\frac{2}{3}$ & 0 \\ - $\sin \cdot \cos^2$ & $\frac{4-\sqrt{2}}{12}$ & $\frac{1}{3}$ & $\frac{2}{3}$ & 0 & 0 & 0 & 0 \\ + $\sin \cdot \cos^2$ & $\frac{4-\sqrt{2}}{12}$ & $\frac{1}{3}$ & $\frac{2}{3}$ & 0 & 0 & 0 & 0 \end{tabularx} + \renewcommand{\arraystretch}{1} + \setlength\tabcolsep{6pt} % default value: 6pt \subsubsection{Intergation of power, rational, exponential and logarithmic functions} diff --git a/src/sections/FourierAnalysis.tex b/src/sections/FourierAnalysis.tex index fa285c9..23dff43 100644 --- a/src/sections/FourierAnalysis.tex +++ b/src/sections/FourierAnalysis.tex @@ -24,7 +24,7 @@ \subsubsection{Properties of Periodic Functions} \begin{enumerate} \item if $f$ is $periodic$ and $continuous$: $f$ is $bounded$ \item if $f$ is $periodic$ and $smooth$: $f$ and $f^{(n)}$ are $bounded$\\ - - smooth means no kinks ("Knicks"), meaning it is infinitely differentiable\\ + - smooth means no kinks (Knicks), meaning it is infinitely differentiable\\ - if $f$ or one of its derivatives is not bounded, we know that $f$ is either not periodic or not smooth. \end{enumerate} @@ -65,7 +65,7 @@ \subsection{Fourier Series} b_{m} & =\frac{1}{L}\int_{-L}^{L}f(x)\sin\left(\frac{m\pi}{L}x\right)dx,\text{ if }m>0 \end{align*} \subsubsection{Dirichlet Theorem - Convergenve} -For discontinuities $f(x^-)\neq f(x^+)$ the fourier series converges to: +For discontinuities $f(x^-)\neq f(x^+)$ the Fourier series converges to: \begin{align*} F(x_0) & =\frac12(f(x_0^-)+f(x_0^+)) \end{align*} @@ -117,8 +117,11 @@ \subsubsection{Properties of Even and Odd Functions} \end{enumerate} \subsubsection{Expansion of FS} -If $f(t)$ is defined on $[0,L]$ the expanded function is $2L$ periodic. This method can be used to solve PDEs. \vspace*{8pt}\\ -\begin{minipage}[t]{0.18\textwidth} +If $f(t)$ is defined on $[0,L]$ the expanded function is $2L$ periodic. This method can be used to solve PDEs. + +\vspace*{8pt} + +\begin{minipage}[t]{0.49\linewidth} \textbf{Copy-paste} expansion\\ \begin{tikzpicture} \begin{axis}[ @@ -166,7 +169,7 @@ \subsubsection{Expansion of FS} \end{axis} \end{tikzpicture} \end{minipage} -\begin{minipage}[t]{0.18\textwidth} +\begin{minipage}[t]{0.49\linewidth} \textbf{Even} expansion\\ \begin{tikzpicture} \begin{axis}[ @@ -214,7 +217,7 @@ \subsubsection{Expansion of FS} \end{axis} \end{tikzpicture} \end{minipage} -\begin{minipage}[t]{0.18\textwidth} +\begin{minipage}[t]{0.49\linewidth} \textbf{Odd} expansion\\ \begin{tikzpicture} \begin{axis}[ @@ -264,7 +267,7 @@ \subsubsection{Expansion of FS} \end{minipage} \subsection{Complex Fourier Series} -If $f(t)$ is $2L$-periodic, the \textbf{complex fourier series} is given by: +If $f(t)$ is $2L$-periodic, the \textbf{complex Fourier series} is given by: \begin{equation*} f(x)=c_0+\sum_{\overset{n=-\infty}{n\neq0}}^\infty c_n\cdot e^{\frac{i\pi n}Lx} \end{equation*} @@ -272,7 +275,8 @@ \subsection{Complex Fourier Series} c_0=\frac1{2L}\int_{-L}^L f(x)dx\quad\mid \quad c_n=\frac1{2L}\int_{-L}^L f(x)\cdot e^{-\frac{i\pi n}Lx}dx \end{equation*} -Conversion from and to \textbf{real FS}:\\ +Conversion from and to \textbf{real FS}: + \begin{tabular}[h]{c|c|c} % chktex -2 \multicolumn{2}{c}{$a_0=c_0$} & $e^{\pm ix}=\cos x\pm i\sin x$ \\ $a_n=c_n+\overline{c_n}$ & $c_n = \frac{1}{2}(a_n-ib_n)$ & $e^{ix}-e^{-ix}=2i\sin x$ \\ @@ -300,7 +304,7 @@ \subsection{Fourier Integral} B(w) & =\frac1\pi\int_{-\infty}^{\infty}f(v)\sin(wv)dv \end{align*} \textbf{Remark}\\ -The convergence properties and behaviour at discontinuities are analogous to the fourier series. +The convergence properties and behavior at discontinuities are analogous to the Fourier series. \subsubsection{Even FI} If $f(t)$ is even, i.e. $f(t)=f(-t)$, the FI simplifies to: \begin{align*} diff --git a/src/sections/FourierTable.tex b/src/sections/FourierTable.tex index 54e839b..bb87839 100644 --- a/src/sections/FourierTable.tex +++ b/src/sections/FourierTable.tex @@ -10,7 +10,7 @@ \subsubsection{Known Transforms}\label{known_transforms} \end{align*} \subsubsection{Fourier transform of time continuous signals} -{\footnotesize \textbf{Remark} The following fourier transformations are taken from \textit{Formelsammlung Signal- und Systemtheorie I} by Prof. Dr. Helmut Bölskei, +{\footnotesize \textbf{Remark} The following Fourier transformations are taken from \textit{Formelsammlung Signal- und Systemtheorie I} by Prof.\ Dr.\ Helmut Bölcskei, where the Fourier Transform is defined without the factor $\frac{1}{\sqrt{2\pi}}$.} \begin{equation*} \widehat{x}(f) =\int_{-\infty}^{\infty} x(t) e^{-2\pi ift}\; dt diff --git a/src/sections/LaplaceTransformation.tex b/src/sections/LaplaceTransformation.tex index 5fa064c..9f2c860 100644 --- a/src/sections/LaplaceTransformation.tex +++ b/src/sections/LaplaceTransformation.tex @@ -21,7 +21,7 @@ \subsubsection{Growth Restriction} \begin{equation*} \left|f(t)\right|\leq Me^{kt} \end{equation*} -for all $t\geq 0, $ then $\mathcal{L}[f](s) $ exists on the interval $(k,\infty).$ +for all $t\geq 0, $ then $\mathcal{L}[f](s) $ exists on the interval $(k,\infty)$. \subsection{Linearity} @@ -65,7 +65,7 @@ \subsection{s-shifting} \end{examplesection} \subsection{Differentiation}\label{ssec:laplaceDiff} -Let $f, f', \dots, f^{(n-1)}$ be continuous functions for all $t \geq 0$. Assume that $f^{(n)}$ is piecewise continuous on every finite subinterval in $[0;1)$. Then % chktex 9 +Let $f, f', \dots, f^{(n-1)}$ be continuous functions for all $t \geq 0$. Assume that $f^{(n)}$ is piecewise continuous on every finite subintervall in $[0;1)$. Then % chktex 9 \begin{equation*} \mathcal{L}[f^{(n)}](s)=s^n\mathcal{L}[f](s)-\sum_{j=0}^{n-1}s^{n-1-j}f^{(j)}(0),\text{ for every }n\geq1 \end{equation*} @@ -313,7 +313,7 @@ \subsection{Convolution} \begin{equation*} (f*g)(t):=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau \end{equation*} -of $f$ and $g$ exists if $f,g\in L^1$ i. e. +of $f$ and $g$ exists if $f,g\in L^1$ i.e. \begin{equation*} \int_{-\infty}^{\infty}\left|f(x)\right|dx<\infty\quad\text{ and }\quad\int_{-\infty}^{\infty}\left|g(x)\right|dx<\infty \end{equation*} diff --git a/src/sections/PartialDifferentialEquations.tex b/src/sections/PartialDifferentialEquations.tex index 8c10055..97bc629 100644 --- a/src/sections/PartialDifferentialEquations.tex +++ b/src/sections/PartialDifferentialEquations.tex @@ -83,7 +83,7 @@ \subsubsection{Classification by Discriminant} \begin{enumerate} \item \textbf{hyperbolic} (wave), if $AC-B^2<0$ \item \textbf{parabolic} (heat), if $AC-B^2=0$ - \item \textbf{elliptic} (laplace), if $AC-B^2>0$ + \item \textbf{elliptic} (Laplace), if $AC-B^2>0$ \end{enumerate} \vspace{5pt} Remarks: @@ -222,7 +222,7 @@ \subsubsection{Inhomogeneous Boundary Conditions} Inserting $F_n$ and $G_n$ into $u(x,t)=F(x)G(t)$ we obtain a family of solutions for our initial problem of the form \[u_n (x, t) = (B_n \cos{(\lambda_{n} t)} + B_n^* \sin{(\lambda_{n}t)})\sin{\left( \frac{n \pi}{L} x \right) } \] where $u_n (x, t)$ are called Eigenfunctions and $\lambda_n$ the spectrum.\\ - Using fourier series we can generate a general solution for all $n$ of the form + Using Fourier series we can generate a general solution for all $n$ of the form \[u(x,t)=\sum\limits_{n=1}^{\infty}\begin{pmatrix} B_n \cos(\lambda _n t)+B_n^* \sin(\lambda_n t) \end{pmatrix} \sin\left(\frac{n \pi}{L}x \right) \] @@ -348,7 +348,7 @@ \subsubsection{Steps} \item Obtain $u(x,y)$ by integrating the \textit{normal form} $u(v,w)$ and re-substitution: \begin{align*} - u(v,w) & =\iint u_{wv}\ dwdv \\ + u(v,w) & =\iint u_{wv}\ dw dv \\ u(x,y) & =u(v,w)|_{v=C_{1}, w=C_{2}} \end{align*} \end{enumerate} @@ -375,7 +375,7 @@ \subsubsection{Boundary Condition \texorpdfstring{$u(0,t), u(L,t)$}{u (0,t), u ( u_x(x,t) & =\sum_{n=1}^\infty B_n \frac{n\pi}L\cos(\frac{n\pi}Lx)e^{-\lambda_n^2t} & \lambda_n = \frac{cn\pi}{L} \\ B_n & =\dfrac{2}{n\pi}\int\limits_0^{L}h(x)\cos(\dfrac{n\pi}{L}x)dx \end{align*} -Assuming $h(x)$ fulfils $\int_{0}^{L}h(x)\;dx = 0$ in order for a solution to exist (see Section \ref{sssec:laplace_neu_rad}) +Assuming $h(x)$ fulfils $\int_{0}^{L}h(x)\;dx = 0$ in order for a solution to exist (see Section~\ref{sssec:laplace_neu_rad}) \subsubsection{Boundary Condition \texorpdfstring{$u_{t}(0,t),u_{t}(L,t)$}{ut (0,t), ut (L,t)}} The heat equation PDE with boundary conditions @@ -614,7 +614,7 @@ \subsubsection{Maximum Principle} \max_{(x,y)\in D_R} u(r,\varphi) & = \max_{\varphi\in[0,2\pi]} u(R,\varphi) \\ \min_{(x,y)\in D_R} u(r,\varphi) & = \min_{\varphi\in[0,2\pi]} u(R,\varphi) \end{align*} -i.e. the maximum of the border $\partial D$ of $D$ is also the maximum of the whole region $D$. +i.e.\ the maximum of the border $\partial D$ of $D$ is also the maximum of the whole region $D$. \subsection{2D Wave Equation - Rectangular Membrane}\label{ssec:2d_wave_rect} For a homogeneous and flexible two-dimensional membrane with time-invariant tension per unit-length, @@ -631,10 +631,10 @@ \subsection{2D Wave Equation - Rectangular Membrane}\label{ssec:2d_wave_rect} \begin{align*} u(x,y,t) & =\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\left[B_{mn}\cos(\lambda_{mn}t)+B_{mn}^{*}\sin(\lambda_{mn}t)\right]\dots \\&\dots \sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right)\\\\ - f(x,y) & =\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}B_{mn}\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right) \\ - B_{mn} & = \frac4{ab}\int_0^b\int_0^{a}f(x,y)\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right)dxdy \\\\ - g(x,y) & =\sum_{m=1}^\infty\sum_{n=1}^\infty B_{mn}^*\lambda_{mn}t\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right) \\ - B_{mn}^{*} & =\frac4{ab\lambda_{mn}}\int_{0}^{b}\int_{0}^{a}g(x,y)\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right)dxdy \\ + f(x,y) & =\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}B_{mn}\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right) \\ + B_{mn} & = \frac4{ab}\int_0^b\int_0^{a}f(x,y)\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right)dx dy \\\\ + g(x,y) & =\sum_{m=1}^\infty\sum_{n=1}^\infty B_{mn}^*\lambda_{mn}t\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right) \\ + B_{mn}^{*} & =\frac4{ab\lambda_{mn}}\int_{0}^{b}\int_{0}^{a}g(x,y)\sin\left(\frac{m\pi x}a\right)\sin\left(\frac{n\pi y}b\right)dx dy \\ \lambda_{mn} & = c\pi\sqrt{\frac{m^2}{a^2}+\frac{n^2}{b^2}} \end{align*} From d08100b04bdc7c4636e1c02b073c8a0c6658b870 Mon Sep 17 00:00:00 2001 From: Tobias Meier Date: Fri, 16 Feb 2024 23:53:15 +0100 Subject: [PATCH 06/13] readme updated --- README.md | 23 +++++++++++------------ 1 file changed, 11 insertions(+), 12 deletions(-) diff --git a/README.md b/README.md index b38abe9..a51a8f8 100644 --- a/README.md +++ b/README.md @@ -7,13 +7,13 @@ It was originally created by Benno Kaeslin, Linard Furck, Sandro Christen, Marku A pre-build pdf file can be downloaded here: -- [Analysis_III_full.pdf](https://meiertobias.github.io/eth-analysis-3/analysis_III_full_portrait.pdf) -- [Analysis_III.pdf (without examples)](https://meiertobias.github.io/eth-analysis-3/analysis_III_portrait.pdf) -- [Analysis_III_full_landscape.pdf](https://meiertobias.github.io/eth-analysis-3/analysis_III_full_landscape.pdf) -- [Analysis_III_landscape.pdf (without examples)](https://meiertobias.github.io/eth-analysis-3/analysis_III_landscape.pdf) +- [Analysis_III_full.pdf](https://meiertobias.github.io/eth-analysis-3/analysis_III_full.pdf) +- [Analysis_III.pdf (without examples)](https://meiertobias.github.io/eth-analysis-3/analysis_III.pdf) ## Getting-Started +To clone this repo and the necessary submodules add the `--recurse-submodules` flag to your `git clone` command. If you forgot to add the `--recurse-submodules` flag when you cloned the repo you can pull the submodules by running `git submodule init` and then `git submodule update` in the root directory of repository. + After you cloned this repo there are two options to build the LaTeX files ether you [use the Docker Container](#docker-recommended) or build it with your [native installed LaTeX compile](#native-latex-compiler) of choice. If you want a hassle-free way to work with our LaTeX project, we recommend using Docker. @@ -42,9 +42,6 @@ Remove unnecessary build files: "latex-workshop.latex.clean.fileTypes" : [ "*.aux", "*.fls", "*.synctex.gz", "*.out", "*.log", "*.fdb_latexmk" ], "latex-workshop.latex.autoClean.run": "onBuilt", "latex-workshop.latex.clean.method": "glob", - -// if using subfolder -"latex-workshop.latex.outDir": "../build", "latex-workshop.latex.clean.subfolder.enabled": true, ``` @@ -57,21 +54,23 @@ To enable the LaTeX checker ChkTeX: To get more information about the different warnings search for your warning code in the ChkTeX [documentation](https://mirror.init7.net/ctan/support/chktex/ChkTeX.pdf). If you want to disable a checker warning on a specific line you can add `% chktex ##` to the end of the line with the warning number you want to suppress. -## Setup on macOS -This repository is configured for Git [LFS](https://git-lfs.com/) (Large File Storage). If you don't already have it installed run `brew install git-lfs` or check the linked website to install it. +## Prerequisites -To use the formatter by LaTeX Workshop install `brew install latexindent`. +The images and other compressed files are managed by git-lfs. Therefor you need to [install git-lfs](https://git-lfs.com/) fist. + +All LaTeX files are autoformatted with `latexindent` which comes within the TexLive installation. If you haven't already install it simply run `brew install latexindent`. ## Python scripts for PDE visualization + Additional Python scripts used for the visualization of various PDEs can be found in [`python_scripts/pde_plot`](python_scripts/pde_plot/). The file [`pde_plot.py`](python_scripts/pde_plot/pde_plot.py) demonstrates how to use the visualization class. ## Contributing Feel free to fork this repository and create a pull request to integrate your corrections and extensions. -## Acknowledgements +## Template -Thanks to *S.* for creating this wonderful AMIV/ETHZ summaries LaTeX template which can be downloaded [here](https://de.overleaf.com/latex/templates/amiv-slash-ethz-summaries-template-landscape/trggddjtjhqr). +The LaTeX template of this summary can be found [here](https://github.com/MeierTobias/eth-summary-template). ## License From 70c5631a853930faa14c16dd914345f3b5cab190 Mon Sep 17 00:00:00 2001 From: Tobias Meier Date: Fri, 16 Feb 2024 23:55:26 +0100 Subject: [PATCH 07/13] images tracked with git-lfs --- .gitattributes | 2 ++ python_scripts/pde_plot/1d_wave.png | Bin 1009233 -> 132 bytes python_scripts/pde_plot/1d_wave_fs.png | Bin 861744 -> 131 bytes python_scripts/pde_plot/2d_laplace.png | Bin 1162511 -> 132 bytes src/images/1d_heat_fs.png | Bin 1266345 -> 132 bytes src/images/1d_wave.png | Bin 1210923 -> 132 bytes src/images/1d_wave_fs.png | Bin 1042647 -> 132 bytes src/images/1d_wave_inhom.png | Bin 1250489 -> 132 bytes src/images/2d_laplace.png | Bin 1255784 -> 132 bytes src/images/dirichlet_superposition.png | Bin 254722 -> 131 bytes src/images/dirichlet_superposition_old.png | Bin 284307 -> 131 bytes src/images/pde_dAlembert.png | Bin 91143 -> 130 bytes src/images/pde_harmonic_function.png | Bin 130008 -> 131 bytes src/images/pde_numerical_1d_heat_eq.png | Bin 4829 -> 129 bytes src/images/pde_numerical_methods_1d_heat.png | Bin 68842 -> 130 bytes src/images/pde_numerical_methods_1d_wave.png | Bin 83556 -> 130 bytes .../pde_numerical_methods_2d_laplace.png | Bin 61901 -> 130 bytes src/images/unit_circle.png | Bin 234596 -> 131 bytes 18 files changed, 2 insertions(+) diff --git a/.gitattributes b/.gitattributes index 65158a2..388cb64 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1,2 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