Combined Analysis

Combined Analysis I and II to one folder in the fs25 folder

fs25/analysis_I_II/II_differentialrechnung_r/II_differentialrechnung_r.tex

@@ -0,0 +1 @@
+\chapter{Differentialrechnung im $\mathbb{R}^{n}$}

fs25/analysis_I_II/II_integration/II_integration.tex

@@ -0,0 +1 @@
+\chapter{Mehrdimensionale Riemann-integration, Satz von Fubini über wiederholte Integration, Jordan-Mass, Substitutionsregel für mehrdimensionale Integrale}

fs25/analysis_I_II/II_stetigkeit_topologie/II_stetigkeit_topologie.tex

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+\chapter{Topologie, Stetigkeit}

fs25/analysis_I_II/I_differentialrechnung_r/I_differentialrechnung_r.tex

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+\chapter{Differentialrechnung auf $\mathbb{R}$}
+
+Intuitiv ist die Ableitung einer Funktion $f : \mathbb{R} \rightarrow \mathbb{R}$ an einer Stelle $x_0 \in \mathbb{R}$ die Steigung der Tangente an den Graphen von $f$ durch den Punkt $x_0, f(x_0)$. Genauer gesagt, ist die Ableitung der Grenzwert der Steigungen der Sekanten durch $(x_0, f(x_0)$ und $(x, f(x))$ für $x$ gegen $x_0$. Ableitungen sind allgegenwärtig in den Wissenschaften und im Ingenieurwesen. In der Mechanik ist die Geschwindigkeit eines Teilchens zum Beispiel die Ableitung seines Ortes als eine Funktion der Zeit. Als ein anderes Beispiel ist in einem elektrischen Schwingkreis die Stromstärke gleich der Ableitung der Ladung des Kondensators als eine Funktion der Zeit.
+
+\input{differential_differentiationsregel.tex}
+\input{mittelwertsatz_folgerungen.tex}
+

fs25/analysis_I_II/I_differentialrechnung_r/differential_differentiationsregel.tex

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+\section{Differential und Differentiationsregeln}
+
+\nt{
+\[
+  dx \text{ und } dy \text{ sind Differentialen. } \Rightarrow f'(x_0) = \frac{dy}{dx}, dx \text{ nennt man Differentialquotient.}
+.\]
+}
+
+\nt{
+  Je kleiner $\Delta x$ ist, desto näher kommt es an den Wert von $\Delta y$ für:
+  \[
+    \Delta y \equiv f'(x_0) \Delta x
+  .\]
+
+  % TODO: Add figure
+}
+
+\exa{}{
+\[
+  f : \mathbb{R} \rightarrow \mathbb{R}, f(x) := x ^2, f'(x_0 = 0) = 0
+.\]
+
+%TODO: Add figure
+}
+
+%TODO: Add proof for Summen-, Produkt- und Quotioentenregel
+
+%TODO: Complete Proof
+
+%TODO: Complete Proof
+
+\nt{
+  Die Menge $U$ ist offen, da $f$ stetig ist. Dies folgt aus dem Fakt, da das Urbild $f ^{-1}(V)$ offen ist $\forall V$ offen. 
+}
+
+%TODO: Add Examples and reference Exercises
+

fs25/analysis_I_II/I_differentialrechnung_r/mittelwertsatz_folgerungen.tex

@@ -0,0 +1,28 @@
+\section{Mittelwertsatz und Folgerungen}
+
+\nt{
+  Für 
+  \[
+    f'(x_0) = \frac{f(b)-f(a)}{b-a}
+  .\]
+
+  gilt, dass $a$ die Steigung der Sekante ist.
+
+  %TODO: Add fig
+}
+
+\nt{
+  $\Rightarrow$ Die Funktion hat genau eine Lösung, die auch nach $f(0) = y_0$ erfüllt, mit $y_0 \in \mathbb{R}$ vorgegeben, nämlich $f(x) = ye ^{cx}$.
+}
+
+\exa{}{
+  \[
+  \frac{x ^{n}}{e ^{(x ^{n})}} ?
+  .\]
+
+  \[
+    \frac{n \cdot x ^{n-1}}{e ^{(x ^{n})}} = \frac{e ^{x} - 1}{x} = \text{exp}(0) = 1
+  .\]
+}
+
+% TODO: Add Proof

fs25/analysis_I_II/I_integration/I_integration.tex

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+\chapter{Integration}

fs25/analysis_I_II/I_stetigkeit_topologie/I_stetigkeit_topologie.tex

@@ -0,0 +1,3 @@
+\chapter{Stetigkeit, Topologie}
+
+\newpage

fs25/analysis_I_II/analysis_I.tex

@@ -0,0 +1,8 @@
+\part{Analysis I}
+
+\import{./grundlagen}{grundlagen.tex}
+\import{./zahlen_und_vektoren}{zahlen_und_vektoren.tex}
+\import{./folgen_und_reihen}{folgen_und_reihen.tex}
+\import{./I_stetigkeit_topologie}{I_stetigkeit_topologie.tex}
+\import{./I_differentialrechnung_r}{I_differentialrechnung_r.tex}
+\import{./I_integration}{I_integration.tex}

fs25/analysis_I_II/analysis_II.tex

@@ -0,0 +1,8 @@
+\part{Analysis II}
+
+\import{./gdg}{gdg.tex}
+\import{./II_stetigkeit_topologie}{II_stetigkeit_topologie.tex}
+\import{./II_differentialrechnung_r}{II_differentialrechnung_r.tex}
+\import{./umkehrsatz}{umkehrsatz.tex}
+\import{./II_integration}{II_integration.tex}
+\import{./vektorfeld}{vektorfeld.tex}

fs25/analysis_I_II/disclaimer.tex

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+\section*{DISCLAIMER}
+Diese Notizen wurden verfasst auf Basis der Vorlesung Analysis I (HS24) von F. Ziltener und dem Skript "Analysis für Informatik" von Michael Struwe.
+\\
+\\
+Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!).
+\\
+\\
+Alle Grafiken wurden eigenhändig mit Manim \cite{MCD2024} generiert.
+\\
+\\
+Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden.

fs25/analysis_I_II/folgen_und_reihen/folgen_und_reihen.tex

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+\chapter{Folgen und Reihen}
+

fs25/analysis_I_II/gdg/gdg.tex

@@ -0,0 +1 @@
+\chapter{Gewöhnliche Differentialgleichungen, Anwendung auf die Mechanik und die Elektrotechnik}

fs25/analysis_I_II/grundlagen/grundlagen.tex

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+\chapter{Grundlagen: Logik, Mengen, Funktionen}
+
+\input{logik.tex}

fs25/analysis_I_II/grundlagen/logik.tex

@@ -0,0 +1,54 @@
+\section{Logik}
+
+In der Logik werden (mathematische) Aussagen untersucht. Eine Aussage ist eine Äusserung, die entweder wahr oder falsch ist. \cite{Ziltner2024} (Wahr oder Falsch).
+\\
+\\
+In der mathematischen Logik gelten die folgenden Sätze.
+
+\begin{itemize}
+	\item \textbf{Satz vom ausgeschlossenen Wiederspruch:} Eine Aussage ist nicht sowohl war als auch falsch.
+	\item \textbf{Satz vom ausgeschlossenen Dritten:} Jede Aussage ist wahr oder falsch.
+\end{itemize} \cite{Ziltner2024}
+
+\nt{
+	Es gibt gewisse Aussagen, als logische Aussage gelten könnte aber nicht zulässig ist. Solche Aussagen sind meisten rückbezügliche Äusserungen und sind deswegen keine sinnvollen Aussagen. (Siege Lügner-Paradox)
+}
+
+Aussagen können verneint und miteinander verknüpft werden.
+
+\begin{center}
+	\begin{tabular}{| c | c | c |}
+		Notation & Bedeutung & Bezeichnung \\
+		\hline
+		T        & wahr      &             \\
+		F        & falsch    &             \\
+		$\neg A$ & nicht A   & Negation    \\
+	\end{tabular}
+\end{center}
+
+Für Verknüpfungen verwenden wir folgende Notationen.
+
+\begin{center}
+	\begin{tabular}{| c | c | c |}
+		Notation              & Bedeutung                & Bezeichnung           \\
+		\hline
+		$A \land B$           & $A$ und $B$              & Konjunktion           \\
+		$A \lor B$            & $A$ oder $B$             & inklusive Disjunktion \\
+		$A \dot{\lor} B$      & entweder $A$ oder $B$    & exklusive Disjunktion \\
+		$A \Rightarrow B$     & wenn $A$, dann $B$       & Implikation           \\
+		$A \Leftrightarrow B$ & genau dann $A$, wenn $B$ & Äquivalenz            \\
+	\end{tabular}
+\end{center}
+
+Die Wahrheitstablelle der vorher erwähnten Verknüpfungen sind wie folgt.
+
+\begin{center}
+	\begin{tabular}{| c | c | c | c | c | c | c |}
+		$A$ & $B$ & $A \land B$ & $A \lor B$ & $A \dot{\lor} B$ & $A \Rightarrow B$ & $A \Leftrightarrow B$ \\
+		\hline
+		F   & F   & F           & F          & F                & T                 & T                     \\
+		F   & T   & F           & T          & T                & T                 & F                     \\
+		T   & F   & F           & T          & T                & F                 & F                     \\
+		T   & T   & T           & T          & F                & T                 & T                     \\
+	\end{tabular}
+\end{center}

fs25/analysis_I_II/letterfonts.tex

@@ -0,0 +1,178 @@
+% Things Lie
+\newcommand{\kb}{\mathfrak b}
+\newcommand{\kg}{\mathfrak g}
+\newcommand{\kh}{\mathfrak h}
+\newcommand{\kn}{\mathfrak n}
+\newcommand{\ku}{\mathfrak u}
+\newcommand{\kz}{\mathfrak z}
+\DeclareMathOperator{\Ext}{Ext} % Ext functor
+\DeclareMathOperator{\Tor}{Tor} % Tor functor
+\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group
+\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6
+
+% More script letters etc.
+\newcommand{\SA}{\mathcal A}
+\newcommand{\SB}{\mathcal B}
+\newcommand{\SC}{\mathcal C}
+\newcommand{\SF}{\mathcal F}
+\newcommand{\SG}{\mathcal G}
+\newcommand{\SH}{\mathcal H}
+\newcommand{\OO}{\mathcal O}
+
+\newcommand{\SCA}{\mathscr A}
+\newcommand{\SCB}{\mathscr B}
+\newcommand{\SCC}{\mathscr C}
+\newcommand{\SCD}{\mathscr D}
+\newcommand{\SCE}{\mathscr E}
+\newcommand{\SCF}{\mathscr F}
+\newcommand{\SCG}{\mathscr G}
+\newcommand{\SCH}{\mathscr H}
+
+% Mathfrak primes
+\newcommand{\km}{\mathfrak m}
+\newcommand{\kp}{\mathfrak p}
+\newcommand{\kq}{\mathfrak q}
+
+% number sets
+\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}}
+\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}}
+\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}}
+\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}}
+\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}}
+\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}}
+\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}}
+\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}}
+% expected value
+\newcommand{\EE}{\ensuremath{\mathbb{E}}}
+\newcommand{\charin}{\text{ char }}
+\DeclareMathOperator{\sign}{sign}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Inn}{Inn}
+\DeclareMathOperator{\Syl}{Syl}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\GL}{GL} % General linear group
+\DeclareMathOperator{\SL}{SL} % Special linear group
+
+%---------------------------------------
+% BlackBoard Math Fonts :-
+%---------------------------------------
+
+%Captital Letters
+\newcommand{\bbA}{\mathbb{A}}	\newcommand{\bbB}{\mathbb{B}}
+\newcommand{\bbC}{\mathbb{C}}	\newcommand{\bbD}{\mathbb{D}}
+\newcommand{\bbE}{\mathbb{E}}	\newcommand{\bbF}{\mathbb{F}}
+\newcommand{\bbG}{\mathbb{G}}	\newcommand{\bbH}{\mathbb{H}}
+\newcommand{\bbI}{\mathbb{I}}	\newcommand{\bbJ}{\mathbb{J}}
+\newcommand{\bbK}{\mathbb{K}}	\newcommand{\bbL}{\mathbb{L}}
+\newcommand{\bbM}{\mathbb{M}}	\newcommand{\bbN}{\mathbb{N}}
+\newcommand{\bbO}{\mathbb{O}}	\newcommand{\bbP}{\mathbb{P}}
+\newcommand{\bbQ}{\mathbb{Q}}	\newcommand{\bbR}{\mathbb{R}}
+\newcommand{\bbS}{\mathbb{S}}	\newcommand{\bbT}{\mathbb{T}}
+\newcommand{\bbU}{\mathbb{U}}	\newcommand{\bbV}{\mathbb{V}}
+\newcommand{\bbW}{\mathbb{W}}	\newcommand{\bbX}{\mathbb{X}}
+\newcommand{\bbY}{\mathbb{Y}}	\newcommand{\bbZ}{\mathbb{Z}}
+
+%---------------------------------------
+% MathCal Fonts :-
+%---------------------------------------
+
+%Captital Letters
+\newcommand{\mcA}{\mathcal{A}}	\newcommand{\mcB}{\mathcal{B}}
+\newcommand{\mcC}{\mathcal{C}}	\newcommand{\mcD}{\mathcal{D}}
+\newcommand{\mcE}{\mathcal{E}}	\newcommand{\mcF}{\mathcal{F}}
+\newcommand{\mcG}{\mathcal{G}}	\newcommand{\mcH}{\mathcal{H}}
+\newcommand{\mcI}{\mathcal{I}}	\newcommand{\mcJ}{\mathcal{J}}
+\newcommand{\mcK}{\mathcal{K}}	\newcommand{\mcL}{\mathcal{L}}
+\newcommand{\mcM}{\mathcal{M}}	\newcommand{\mcN}{\mathcal{N}}
+\newcommand{\mcO}{\mathcal{O}}	\newcommand{\mcP}{\mathcal{P}}
+\newcommand{\mcQ}{\mathcal{Q}}	\newcommand{\mcR}{\mathcal{R}}
+\newcommand{\mcS}{\mathcal{S}}	\newcommand{\mcT}{\mathcal{T}}
+\newcommand{\mcU}{\mathcal{U}}	\newcommand{\mcV}{\mathcal{V}}
+\newcommand{\mcW}{\mathcal{W}}	\newcommand{\mcX}{\mathcal{X}}
+\newcommand{\mcY}{\mathcal{Y}}	\newcommand{\mcZ}{\mathcal{Z}}
+
+
+%---------------------------------------
+% Bold Math Fonts :-
+%---------------------------------------
+
+%Captital Letters
+\newcommand{\bmA}{\boldsymbol{A}}	\newcommand{\bmB}{\boldsymbol{B}}
+\newcommand{\bmC}{\boldsymbol{C}}	\newcommand{\bmD}{\boldsymbol{D}}
+\newcommand{\bmE}{\boldsymbol{E}}	\newcommand{\bmF}{\boldsymbol{F}}
+\newcommand{\bmG}{\boldsymbol{G}}	\newcommand{\bmH}{\boldsymbol{H}}
+\newcommand{\bmI}{\boldsymbol{I}}	\newcommand{\bmJ}{\boldsymbol{J}}
+\newcommand{\bmK}{\boldsymbol{K}}	\newcommand{\bmL}{\boldsymbol{L}}
+\newcommand{\bmM}{\boldsymbol{M}}	\newcommand{\bmN}{\boldsymbol{N}}
+\newcommand{\bmO}{\boldsymbol{O}}	\newcommand{\bmP}{\boldsymbol{P}}
+\newcommand{\bmQ}{\boldsymbol{Q}}	\newcommand{\bmR}{\boldsymbol{R}}
+\newcommand{\bmS}{\boldsymbol{S}}	\newcommand{\bmT}{\boldsymbol{T}}
+\newcommand{\bmU}{\boldsymbol{U}}	\newcommand{\bmV}{\boldsymbol{V}}
+\newcommand{\bmW}{\boldsymbol{W}}	\newcommand{\bmX}{\boldsymbol{X}}
+\newcommand{\bmY}{\boldsymbol{Y}}	\newcommand{\bmZ}{\boldsymbol{Z}}
+%Small Letters
+\newcommand{\bma}{\boldsymbol{a}}	\newcommand{\bmb}{\boldsymbol{b}}
+\newcommand{\bmc}{\boldsymbol{c}}	\newcommand{\bmd}{\boldsymbol{d}}
+\newcommand{\bme}{\boldsymbol{e}}	\newcommand{\bmf}{\boldsymbol{f}}
+\newcommand{\bmg}{\boldsymbol{g}}	\newcommand{\bmh}{\boldsymbol{h}}
+\newcommand{\bmi}{\boldsymbol{i}}	\newcommand{\bmj}{\boldsymbol{j}}
+\newcommand{\bmk}{\boldsymbol{k}}	\newcommand{\bml}{\boldsymbol{l}}
+\newcommand{\bmm}{\boldsymbol{m}}	\newcommand{\bmn}{\boldsymbol{n}}
+\newcommand{\bmo}{\boldsymbol{o}}	\newcommand{\bmp}{\boldsymbol{p}}
+\newcommand{\bmq}{\boldsymbol{q}}	\newcommand{\bmr}{\boldsymbol{r}}
+\newcommand{\bms}{\boldsymbol{s}}	\newcommand{\bmt}{\boldsymbol{t}}
+\newcommand{\bmu}{\boldsymbol{u}}	\newcommand{\bmv}{\boldsymbol{v}}
+\newcommand{\bmw}{\boldsymbol{w}}	\newcommand{\bmx}{\boldsymbol{x}}
+\newcommand{\bmy}{\boldsymbol{y}}	\newcommand{\bmz}{\boldsymbol{z}}
+
+%---------------------------------------
+% Scr Math Fonts :-
+%---------------------------------------
+
+\newcommand{\sA}{{\mathscr{A}}}   \newcommand{\sB}{{\mathscr{B}}}
+\newcommand{\sC}{{\mathscr{C}}}   \newcommand{\sD}{{\mathscr{D}}}
+\newcommand{\sE}{{\mathscr{E}}}   \newcommand{\sF}{{\mathscr{F}}}
+\newcommand{\sG}{{\mathscr{G}}}   \newcommand{\sH}{{\mathscr{H}}}
+\newcommand{\sI}{{\mathscr{I}}}   \newcommand{\sJ}{{\mathscr{J}}}
+\newcommand{\sK}{{\mathscr{K}}}   \newcommand{\sL}{{\mathscr{L}}}
+\newcommand{\sM}{{\mathscr{M}}}   \newcommand{\sN}{{\mathscr{N}}}
+\newcommand{\sO}{{\mathscr{O}}}   \newcommand{\sP}{{\mathscr{P}}}
+\newcommand{\sQ}{{\mathscr{Q}}}   \newcommand{\sR}{{\mathscr{R}}}
+\newcommand{\sS}{{\mathscr{S}}}   \newcommand{\sT}{{\mathscr{T}}}
+\newcommand{\sU}{{\mathscr{U}}}   \newcommand{\sV}{{\mathscr{V}}}
+\newcommand{\sW}{{\mathscr{W}}}   \newcommand{\sX}{{\mathscr{X}}}
+\newcommand{\sY}{{\mathscr{Y}}}   \newcommand{\sZ}{{\mathscr{Z}}}
+
+
+%---------------------------------------
+% Math Fraktur Font
+%---------------------------------------
+
+%Captital Letters
+\newcommand{\mfA}{\mathfrak{A}}	\newcommand{\mfB}{\mathfrak{B}}
+\newcommand{\mfC}{\mathfrak{C}}	\newcommand{\mfD}{\mathfrak{D}}
+\newcommand{\mfE}{\mathfrak{E}}	\newcommand{\mfF}{\mathfrak{F}}
+\newcommand{\mfG}{\mathfrak{G}}	\newcommand{\mfH}{\mathfrak{H}}
+\newcommand{\mfI}{\mathfrak{I}}	\newcommand{\mfJ}{\mathfrak{J}}
+\newcommand{\mfK}{\mathfrak{K}}	\newcommand{\mfL}{\mathfrak{L}}
+\newcommand{\mfM}{\mathfrak{M}}	\newcommand{\mfN}{\mathfrak{N}}
+\newcommand{\mfO}{\mathfrak{O}}	\newcommand{\mfP}{\mathfrak{P}}
+\newcommand{\mfQ}{\mathfrak{Q}}	\newcommand{\mfR}{\mathfrak{R}}
+\newcommand{\mfS}{\mathfrak{S}}	\newcommand{\mfT}{\mathfrak{T}}
+\newcommand{\mfU}{\mathfrak{U}}	\newcommand{\mfV}{\mathfrak{V}}
+\newcommand{\mfW}{\mathfrak{W}}	\newcommand{\mfX}{\mathfrak{X}}
+\newcommand{\mfY}{\mathfrak{Y}}	\newcommand{\mfZ}{\mathfrak{Z}}
+%Small Letters
+\newcommand{\mfa}{\mathfrak{a}}	\newcommand{\mfb}{\mathfrak{b}}
+\newcommand{\mfc}{\mathfrak{c}}	\newcommand{\mfd}{\mathfrak{d}}
+\newcommand{\mfe}{\mathfrak{e}}	\newcommand{\mff}{\mathfrak{f}}
+\newcommand{\mfg}{\mathfrak{g}}	\newcommand{\mfh}{\mathfrak{h}}
+\newcommand{\mfi}{\mathfrak{i}}	\newcommand{\mfj}{\mathfrak{j}}
+\newcommand{\mfk}{\mathfrak{k}}	\newcommand{\mfl}{\mathfrak{l}}
+\newcommand{\mfm}{\mathfrak{m}}	\newcommand{\mfn}{\mathfrak{n}}
+\newcommand{\mfo}{\mathfrak{o}}	\newcommand{\mfp}{\mathfrak{p}}
+\newcommand{\mfq}{\mathfrak{q}}	\newcommand{\mfr}{\mathfrak{r}}
+\newcommand{\mfs}{\mathfrak{s}}	\newcommand{\mft}{\mathfrak{t}}
+\newcommand{\mfu}{\mathfrak{u}}	\newcommand{\mfv}{\mathfrak{v}}
+\newcommand{\mfw}{\mathfrak{w}}	\newcommand{\mfx}{\mathfrak{x}}
+\newcommand{\mfy}{\mathfrak{y}}	\newcommand{\mfz}{\mathfrak{z}}

fs25/analysis_I_II/macros.tex

@@ -0,0 +1,88 @@
+%From M275 "Topology" at SJSU
+\newcommand{\id}{\mathrm{id}}
+\newcommand{\taking}[1]{\xrightarrow{#1}}
+\newcommand{\inv}{^{-1}}
+
+%From M170 "Introduction to Graph Theory" at SJSU
+\DeclareMathOperator{\diam}{diam}
+\DeclareMathOperator{\ord}{ord}
+\newcommand{\defeq}{\overset{\mathrm{def}}{=}}
+
+%From the USAMO .tex files
+\newcommand{\ts}{\textsuperscript}
+\newcommand{\dg}{^\circ}
+\newcommand{\ii}{\item}
+
+% % From Math 55 and Math 145 at Harvard
+% \newenvironment{subproof}[1][Proof]{%
+% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}%
+% {\end{proof}}
+
+\newcommand{\liff}{\leftrightarrow}
+\newcommand{\lthen}{\rightarrow}
+\newcommand{\opname}{\operatorname}
+\newcommand{\surjto}{\twoheadrightarrow}
+\newcommand{\injto}{\hookrightarrow}
+\newcommand{\On}{\mathrm{On}} % ordinals
+\DeclareMathOperator{\img}{im} % Image
+\DeclareMathOperator{\Img}{Im} % Image
+\DeclareMathOperator{\coker}{coker} % Cokernel
+\DeclareMathOperator{\Coker}{Coker} % Cokernel
+\DeclareMathOperator{\Ker}{Ker} % Kernel
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\Spec}{Spec} % spectrum
+\DeclareMathOperator{\Tr}{Tr} % trace
+\DeclareMathOperator{\pr}{pr} % projection
+\DeclareMathOperator{\ext}{ext} % extension
+\DeclareMathOperator{\pred}{pred} % predecessor
+\DeclareMathOperator{\dom}{dom} % domain
+\DeclareMathOperator{\ran}{ran} % range
+\DeclareMathOperator{\Hom}{Hom} % homomorphism
+\DeclareMathOperator{\Mor}{Mor} % morphisms
+\DeclareMathOperator{\End}{End} % endomorphism
+
+\newcommand{\eps}{\epsilon}
+\newcommand{\veps}{\varepsilon}
+\newcommand{\ol}{\overline}
+\newcommand{\ul}{\underline}
+\newcommand{\wt}{\widetilde}
+\newcommand{\wh}{\widehat}
+\newcommand{\vocab}[1]{\textbf{\color{blue} #1}}
+\providecommand{\half}{\frac{1}{2}}
+\newcommand{\dang}{\measuredangle} %% Directed angle
+\newcommand{\ray}[1]{\overrightarrow{#1}}
+\newcommand{\seg}[1]{\overline{#1}}
+\newcommand{\arc}[1]{\wideparen{#1}}
+\DeclareMathOperator{\cis}{cis}
+\DeclareMathOperator*{\lcm}{lcm}
+\DeclareMathOperator*{\argmin}{arg min}
+\DeclareMathOperator*{\argmax}{arg max}
+\newcommand{\cycsum}{\sum_{\mathrm{cyc}}}
+\newcommand{\symsum}{\sum_{\mathrm{sym}}}
+\newcommand{\cycprod}{\prod_{\mathrm{cyc}}}
+\newcommand{\symprod}{\prod_{\mathrm{sym}}}
+\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}}
+\newcommand{\parinn}{\setlength{\parindent}{1cm}}
+\newcommand{\parinf}{\setlength{\parindent}{0cm}}
+% \newcommand{\norm}{\|\cdot\|}
+\newcommand{\inorm}{\norm_{\infty}}
+\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}}
+\newcommand{\oset}{V_{\alpha}}
+\newcommand{\opset}[1]{V_{\alpha_{#1}}}
+\newcommand{\lub}{\text{lub}}
+\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}}
+\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}}
+\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}}
+\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}}
+\newcommand{\lm}{\lambda}
+\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}}
+\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}}
+\newcommand{\lt}{\left}
+\newcommand{\rt}{\right}
+\newcommand{\bs}[1]{\boldsymbol{#1}}
+\newcommand{\exs}{\exists}
+\newcommand{\st}{\strut}
+\newcommand{\dps}[1]{\displaystyle{#1}}
+
+\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} }
+\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed}

fs25/analysis_I_II/notizen_analysis_ruh_jirayu.bbl

@@ -0,0 +1,11 @@
+\begin{thebibliography}{}
+
+\bibitem[{The Manim Community Developers}, 2024]{MCD2024}
+{The Manim Community Developers} (2024).
+\newblock {Manim - Mathematical Animation Framework}.
+
+\bibitem[Ziltner, 2024]{Ziltner2024}
+Ziltner, Prof.~Dr., F. (2024).
+\newblock {Notizen zur Vorlesung Analysis 1 für ITET und RW, Herbstsemester 2024}.
+
+\end{thebibliography}

fs25/analysis_I_II/notizen_analysis_ruh_jirayu.tex

@@ -0,0 +1,43 @@
+\documentclass{report}
+
+\def\papertitle{Analysis I}
+
+\def\theorytitle{Theorie}
+\def\corollarytitle{Daraus folgt}
+\def\proposaltitle{Vorschlag}
+\def\claimtitle{Behauptung}
+\def\exercisetitle{Aufgabe}
+\def\exampletitle{Beispiel}
+\def\questiontitle{Frage}
+\def\wrongctitle{Falscher Konzept}
+
+\input{preamble}
+\input{macros}
+\input{letterfonts}
+
+\title{\huge{\papertitle}}
+\author{\huge{Jirayu Ruh}}
+\date{}
+
+\begin{document}
+
+\maketitle
+\newpage% or \cleardoublepage
+% \pdfbookmark[<level>]{<title>}{<dest>}
+\pdfbookmark[section]{\contentsname}{toc}
+\tableofcontents
+\pagebreak
+
+\import{./}{disclaimer.tex}
+\import{./}{analysis_I.tex}
+\import{./}{analysis_II.tex}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% BIBLIOGRAPHY
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\addcontentsline{toc}{chapter}{References}
+\bibliographystyle{apalike}
+\bibliography{sources}
+
+\end{document}

fs25/analysis_I_II/preamble.tex

@@ -0,0 +1,778 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% PACKAGE IMPORTS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\usepackage[german]{babel}
+\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry}
+\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools}
+\usepackage[varbb]{newpxmath}
+\usepackage{xfrac}
+\usepackage[makeroom]{cancel}
+\usepackage{mathtools}
+\usepackage{bookmark}
+\usepackage{enumitem}
+\usepackage{hyperref,theoremref}
+\hypersetup{
+	pdftitle={Assignment},
+	colorlinks=true, linkcolor=doc!90,
+	bookmarksnumbered=true,
+	bookmarksopen=true
+}
+\usepackage[most,many,breakable]{tcolorbox}
+\usepackage{xcolor}
+\usepackage{varwidth}
+\usepackage{varwidth}
+\usepackage{etoolbox}
+%\usepackage{authblk}
+\usepackage{nameref}
+\usepackage{multicol,array}
+\usepackage{tikz-cd}
+\usepackage[ruled,vlined,linesnumbered]{algorithm2e}
+\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) 
+\usepackage{import}
+\usepackage{xifthen}
+\usepackage{pdfpages}
+\usepackage{transparent}
+
+\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}}
+\SetCommentSty{mycommfont}
+\newcommand{\incfig}[1]{%
+    \def\svgwidth{\columnwidth}
+    \import{./figures/}{#1.pdf_tex}
+}
+
+\usepackage{tikzsymbols}
+\renewcommand\qedsymbol{$\Laughey$}
+
+
+%\usepackage{import}
+%\usepackage{xifthen}
+%\usepackage{pdfpages}
+%\usepackage{transparent}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% SELF MADE COLORS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+\definecolor{myg}{RGB}{56, 140, 70}
+\definecolor{myb}{RGB}{45, 111, 177}
+\definecolor{myr}{RGB}{199, 68, 64}
+\definecolor{mytheorembg}{HTML}{F2F2F9}
+\definecolor{mytheoremfr}{HTML}{00007B}
+\definecolor{mylenmabg}{HTML}{FFFAF8}
+\definecolor{mylenmafr}{HTML}{983b0f}
+\definecolor{mypropbg}{HTML}{f2fbfc}
+\definecolor{mypropfr}{HTML}{191971}
+\definecolor{myexamplebg}{HTML}{F2FBF8}
+\definecolor{myexamplefr}{HTML}{88D6D1}
+\definecolor{myexampleti}{HTML}{2A7F7F}
+\definecolor{mydefinitbg}{HTML}{E5E5FF}
+\definecolor{mydefinitfr}{HTML}{3F3FA3}
+\definecolor{notesgreen}{RGB}{0,162,0}
+\definecolor{myp}{RGB}{197, 92, 212}
+\definecolor{mygr}{HTML}{2C3338}
+\definecolor{myred}{RGB}{127,0,0}
+\definecolor{myyellow}{RGB}{169,121,69}
+\definecolor{myexercisebg}{HTML}{F2FBF8}
+\definecolor{myexercisefg}{HTML}{88D6D1}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% TCOLORBOX SETUPS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\setlength{\parindent}{0cm}
+%================================
+% THEOREM BOX
+%================================
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=section]{Theorem}{\theorytitle}
+{%
+	enhanced,
+	breakable,
+	colback = mytheorembg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{mytheoremfr},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = mytheoremfr,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, mytheoremfr},
+}
+{th}
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=chapter]{theorem}{\theorytitle}
+{%
+	enhanced,
+	breakable,
+	colback = mytheorembg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{mytheoremfr},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = mytheoremfr,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, mytheoremfr},
+}
+{th}
+
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcolorbox{Theoremcon}
+{%
+	enhanced
+	,breakable
+	,colback = mytheorembg
+	,frame hidden
+	,boxrule = 0sp
+	,borderline west = {2pt}{0pt}{mytheoremfr}
+	,sharp corners
+	,description font = \mdseries
+	,separator sign none
+}
+
+%================================
+% Corollery
+%================================
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=section]{Corollary}{\corollarytitle}
+{%
+	enhanced
+	,breakable
+	,colback = myp!10
+	,frame hidden
+	,boxrule = 0sp
+	,borderline west = {2pt}{0pt}{myp!85!black}
+	,sharp corners
+	,detach title
+	,before upper = \tcbtitle\par\smallskip
+	,coltitle = myp!85!black
+	,fonttitle = \bfseries\sffamily
+	,description font = \mdseries
+	,separator sign none
+	,segmentation style={solid, myp!85!black}
+}
+{th}
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle}
+{%
+	enhanced
+	,breakable
+	,colback = myp!10
+	,frame hidden
+	,boxrule = 0sp
+	,borderline west = {2pt}{0pt}{myp!85!black}
+	,sharp corners
+	,detach title
+	,before upper = \tcbtitle\par\smallskip
+	,coltitle = myp!85!black
+	,fonttitle = \bfseries\sffamily
+	,description font = \mdseries
+	,separator sign none
+	,segmentation style={solid, myp!85!black}
+}
+{th}
+
+
+%================================
+% LENMA
+%================================
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=section]{Lenma}{Lenma}
+{%
+	enhanced,
+	breakable,
+	colback = mylenmabg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{mylenmafr},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = mylenmafr,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, mylenmafr},
+}
+{th}
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=chapter]{lenma}{Lenma}
+{%
+	enhanced,
+	breakable,
+	colback = mylenmabg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{mylenmafr},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = mylenmafr,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, mylenmafr},
+}
+{th}
+
+
+%================================
+% PROPOSITION
+%================================
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=section]{Prop}{\proposaltitle}
+{%
+	enhanced,
+	breakable,
+	colback = mypropbg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{mypropfr},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = mypropfr,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, mypropfr},
+}
+{th}
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=chapter]{prop}{\proposaltitle}
+{%
+	enhanced,
+	breakable,
+	colback = mypropbg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{mypropfr},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = mypropfr,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, mypropfr},
+}
+{th}
+
+
+%================================
+% CLAIM
+%================================
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=section]{claim}{\claimtitle}
+{%
+	enhanced
+	,breakable
+	,colback = myg!10
+	,frame hidden
+	,boxrule = 0sp
+	,borderline west = {2pt}{0pt}{myg}
+	,sharp corners
+	,detach title
+	,before upper = \tcbtitle\par\smallskip
+	,coltitle = myg!85!black
+	,fonttitle = \bfseries\sffamily
+	,description font = \mdseries
+	,separator sign none
+	,segmentation style={solid, myg!85!black}
+}
+{th}
+
+
+
+%================================
+% Exercise
+%================================
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=section]{Exercise}{\exercisetitle}
+{%
+	enhanced,
+	breakable,
+	colback = myexercisebg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{myexercisefg},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = myexercisefg,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, myexercisefg},
+}
+{th}
+
+\tcbuselibrary{theorems,skins,hooks}
+\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle}
+{%
+	enhanced,
+	breakable,
+	colback = myexercisebg,
+	frame hidden,
+	boxrule = 0sp,
+	borderline west = {2pt}{0pt}{myexercisefg},
+	sharp corners,
+	detach title,
+	before upper = \tcbtitle\par\smallskip,
+	coltitle = myexercisefg,
+	fonttitle = \bfseries\sffamily,
+	description font = \mdseries,
+	separator sign none,
+	segmentation style={solid, myexercisefg},
+}
+{th}
+
+%================================
+% EXAMPLE BOX
+%================================
+
+\newtcbtheorem[number within=section]{Example}{\exampletitle}
+{%
+	colback = myexamplebg
+	,breakable
+	,colframe = myexamplefr
+	,coltitle = myexampleti
+	,boxrule = 1pt
+	,sharp corners
+	,detach title
+	,before upper=\tcbtitle\par\smallskip
+	,fonttitle = \bfseries
+	,description font = \mdseries
+	,separator sign none
+	,description delimiters parenthesis
+}
+{ex}
+
+\newtcbtheorem[number within=chapter]{example}{\exampletitle}
+{%
+	colback = myexamplebg
+	,breakable
+	,colframe = myexamplefr
+	,coltitle = myexampleti
+	,boxrule = 1pt
+	,sharp corners
+	,detach title
+	,before upper=\tcbtitle\par\smallskip
+	,fonttitle = \bfseries
+	,description font = \mdseries
+	,separator sign none
+	,description delimiters parenthesis
+}
+{ex}
+
+%================================
+% DEFINITION BOX
+%================================
+
+\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced,
+	before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm,
+	attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm,
+	boxed title style={frame code={
+					\path[fill=tcbcolback]
+					([yshift=-1mm,xshift=-1mm]frame.north west)
+					arc[start angle=0,end angle=180,radius=1mm]
+					([yshift=-1mm,xshift=1mm]frame.north east)
+					arc[start angle=180,end angle=0,radius=1mm];
+					\path[left color=tcbcolback!60!black,right color=tcbcolback!60!black,
+						middle color=tcbcolback!80!black]
+					([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east)
+					[rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east)
+					-- (frame.south east) -- (frame.south west)
+					-- ([xshift=-1mm,yshift=-1mm]frame.north west)
+					[sharp corners]-- cycle;
+				},interior engine=empty,
+		},
+	fonttitle=\bfseries,
+	title={#2},#1}{def}
+\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced,
+	before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm,
+	attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm,
+	boxed title style={frame code={
+					\path[fill=tcbcolback]
+					([yshift=-1mm,xshift=-1mm]frame.north west)
+					arc[start angle=0,end angle=180,radius=1mm]
+					([yshift=-1mm,xshift=1mm]frame.north east)
+					arc[start angle=180,end angle=0,radius=1mm];
+					\path[left color=tcbcolback!60!black,right color=tcbcolback!60!black,
+						middle color=tcbcolback!80!black]
+					([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east)
+					[rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east)
+					-- (frame.south east) -- (frame.south west)
+					-- ([xshift=-1mm,yshift=-1mm]frame.north west)
+					[sharp corners]-- cycle;
+				},interior engine=empty,
+		},
+	fonttitle=\bfseries,
+	title={#2},#1}{def}
+
+
+
+%================================
+% Solution BOX
+%================================
+
+\makeatletter
+\newtcbtheorem{question}{\questiontitle}{enhanced,
+	breakable,
+	colback=white,
+	colframe=myb!80!black,
+	attach boxed title to top left={yshift*=-\tcboxedtitleheight},
+	fonttitle=\bfseries,
+	title={#2},
+	boxed title size=title,
+	boxed title style={%
+			sharp corners,
+			rounded corners=northwest,
+			colback=tcbcolframe,
+			boxrule=0pt,
+		},
+	underlay boxed title={%
+			\path[fill=tcbcolframe] (title.south west)--(title.south east)
+			to[out=0, in=180] ([xshift=5mm]title.east)--
+			(title.center-|frame.east)
+			[rounded corners=\kvtcb@arc] |-
+			(frame.north) -| cycle;
+		},
+	#1
+}{def}
+\makeatother
+
+%================================
+% SOLUTION BOX
+%================================
+
+\makeatletter
+\newtcolorbox{solution}{enhanced,
+	breakable,
+	colback=white,
+	colframe=myg!80!black,
+	attach boxed title to top left={yshift*=-\tcboxedtitleheight},
+	title=Solution,
+	boxed title size=title,
+	boxed title style={%
+			sharp corners,
+			rounded corners=northwest,
+			colback=tcbcolframe,
+			boxrule=0pt,
+		},
+	underlay boxed title={%
+			\path[fill=tcbcolframe] (title.south west)--(title.south east)
+			to[out=0, in=180] ([xshift=5mm]title.east)--
+			(title.center-|frame.east)
+			[rounded corners=\kvtcb@arc] |-
+			(frame.north) -| cycle;
+		},
+}
+\makeatother
+
+%================================
+% Question BOX
+%================================
+
+\makeatletter
+\newtcbtheorem{qstion}{\questiontitle}{enhanced,
+	breakable,
+	colback=white,
+	colframe=mygr,
+	attach boxed title to top left={yshift*=-\tcboxedtitleheight},
+	fonttitle=\bfseries,
+	title={#2},
+	boxed title size=title,
+	boxed title style={%
+			sharp corners,
+			rounded corners=northwest,
+			colback=tcbcolframe,
+			boxrule=0pt,
+		},
+	underlay boxed title={%
+			\path[fill=tcbcolframe] (title.south west)--(title.south east)
+			to[out=0, in=180] ([xshift=5mm]title.east)--
+			(title.center-|frame.east)
+			[rounded corners=\kvtcb@arc] |-
+			(frame.north) -| cycle;
+		},
+	#1
+}{def}
+\makeatother
+
+\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{
+	breakable,
+	enhanced,
+	colback=white,
+	colframe=myr,
+	arc=0pt,
+	outer arc=0pt,
+	fonttitle=\bfseries\sffamily\large,
+	colbacktitle=myr,
+	attach boxed title to top left={},
+	boxed title style={
+			enhanced,
+			skin=enhancedfirst jigsaw,
+			arc=3pt,
+			bottom=0pt,
+			interior style={fill=myr}
+		},
+	#1
+}{def}
+
+
+
+%================================
+% NOTE BOX
+%================================
+
+\usetikzlibrary{arrows,calc,shadows.blur}
+\tcbuselibrary{skins}
+\newtcolorbox{note}[1][]{%
+	enhanced jigsaw,
+	colback=gray!20!white,%
+	colframe=gray!80!black,
+	size=small,
+	boxrule=1pt,
+	title=\textbf{Bemerkung:-},
+	halign title=flush center,
+	coltitle=black,
+	breakable,
+	drop shadow=black!50!white,
+	attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2},
+	minipage boxed title=2.5cm,
+	boxed title style={%
+			colback=white,
+			size=fbox,
+			boxrule=1pt,
+			boxsep=2pt,
+			underlay={%
+					\coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$);
+					\coordinate (dotB) at ($(interior.east) + (0.5pt,0)$);
+					\begin{scope}
+						\clip (interior.north west) rectangle ([xshift=3ex]interior.east);
+						\filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east);
+					\end{scope}
+					\begin{scope}[gray!80!black]
+						\fill (dotA) circle (2pt);
+						\fill (dotB) circle (2pt);
+					\end{scope}
+				},
+		},
+	#1,
+}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% SELF MADE COMMANDS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}}
+\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}}
+\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}}
+\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}}
+\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}}
+\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}}
+\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}}
+\newcommand{\exe}[2]{\begin{Exercise}{#1}{}#2\end{Exercise}}
+\newcommand{\exa}[2]{\begin{Example}{#1}{}#2\end{Example}}
+\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}}
+\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}}
+\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}}
+\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}}
+\newcommand{\nt}[1]{\begin{note}#1\end{note}}
+
+\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
+	Wrong Concept	\node[shape=circle,draw,inner sep=1pt] (char) {#1};}}
+\newcommand\getcurrentref[1]{%
+	\ifnumequal{\value{#1}}{0}
+	{??}
+	{\the\value{#1}}%
+}
+\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}}
+\newenvironment{myproof}[1][\proofname]{%
+	\proof[\bfseries #1: ]%
+}{\endproof}
+
+\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}}
+\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{}
+
+\newcounter{mylabelcounter}
+
+\makeatletter
+\newcommand{\setword}[2]{%
+	\phantomsection
+	#1\def\@currentlabel{\unexpanded{#1}}\label{#2}%
+}
+\makeatother
+
+
+
+
+\tikzset{
+	symbol/.style={
+			draw=none,
+			every to/.append style={
+					edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
+		}
+}
+
+
+% deliminators
+\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
+\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
+
+\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
+\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
+\DeclarePairedDelimiter{\round}{\lfloor}{\rceil}
+
+\newsavebox\diffdbox
+\newcommand{\slantedromand}{{\mathpalette\makesl{d}}}
+\newcommand{\makesl}[2]{%
+\begingroup
+\sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}%
+\pdfsave
+\pdfsetmatrix{1 0 0.2 1}%
+\rlap{\usebox{\diffdbox}}%
+\pdfrestore
+\hskip\wd\diffdbox
+\endgroup
+}
+\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{%
+\slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}%
+{\slantedromand\hspace{0.2ex}^{#1}}}}
+\ProvideDocumentCommand\dv{o m g}{%
+  \ensuremath{%
+    \IfValueTF{#3}{%
+      \IfNoValueTF{#1}{%
+        \frac{\dd #2}{\dd #3}%
+      }{%
+        \frac{\dd^{#1} #2}{\dd #3^{#1}}%
+      }%
+    }{%
+      \IfNoValueTF{#1}{%
+        \frac{\dd}{\dd #2}%
+      }{%
+        \frac{\dd^{#1}}{\dd #2^{#1}}%
+      }%
+    }%
+  }%
+}
+\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}}
+%  - others
+\DeclareMathOperator{\Lap}{\mathcal{L}}
+\DeclareMathOperator{\Var}{Var} % varience
+\DeclareMathOperator{\Cov}{Cov} % covarience
+\DeclareMathOperator{\E}{E} % expected
+
+% Since the amsthm package isn't loaded
+
+% I prefer the slanted \leq
+\let\oldleq\leq % save them in case they're every wanted
+\let\oldgeq\geq
+\renewcommand{\leq}{\leqslant}
+\renewcommand{\geq}{\geqslant}
+
+% % redefine matrix env to allow for alignment, use r as default
+% \renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep
+%     \let\@ifnextchar\new@ifnextchar
+%     \array{*\c@MaxMatrixCols #1}}
+
+
+%\usepackage{framed}
+%\usepackage{titletoc}
+%\usepackage{etoolbox}
+%\usepackage{lmodern}
+
+
+%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{}
+
+%\renewenvironment{leftbar}
+%{\def\FrameCommand{\hspace{6em}%
+%		{\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}%
+%	\MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt%
+%}
+%{\endMakeFramed}
+
+%\titlecontents{chapter}
+%[0em]{\vspace*{2\baselineskip}}
+%{\parbox{4.5em}{%
+%		\hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}%
+%	\vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily}
+%{}{\endleftbar}
+%\titlecontents{section}
+%[8.4em]
+%{\sffamily\contentslabel{3em}}{}{}
+%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage}
+%\titlecontents{subsection}
+%[8.4em]
+%{\sffamily\contentslabel{3em}}{}{}  
+%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% TABLE OF CONTENTS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\usepackage{tikz}
+\definecolor{doc}{RGB}{0,60,110}
+\usepackage{titletoc}
+\contentsmargin{0cm}
+\titlecontents{chapter}[3.7pc]
+{\addvspace{30pt}%
+	\begin{tikzpicture}[remember picture, overlay]%
+		\draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);%
+		\pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Kapitel\ \thecontentslabel};%
+	\end{tikzpicture}\color{doc!60}\large\sc\bfseries}%
+{}
+{}
+{\;\titlerule\;\large\sc\bfseries Seite \thecontentspage
+	\begin{tikzpicture}[remember picture, overlay]
+		\draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt);
+	\end{tikzpicture}}%
+\titlecontents{section}[3.7pc]
+{\addvspace{2pt}}
+{\contentslabel[\thecontentslabel]{2pc}}
+{}
+{\hfill\small \thecontentspage}
+[]
+\titlecontents*{subsection}[3.7pc]
+{\addvspace{-1pt}\small}
+{}
+{}
+{\ --- \small\thecontentspage}
+[ \textbullet\ ][]
+
+\makeatletter
+\renewcommand{\tableofcontents}{%
+	\chapter*{%
+	  \vspace*{-20\p@}%
+	  \begin{tikzpicture}[remember picture, overlay]%
+		  \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};%
+		  \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);%
+		  \clip (13,-.75) rectangle (20,1);
+		  \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};%
+	  \end{tikzpicture}}%
+	\@starttoc{toc}}
+\makeatother

fs25/analysis_I_II/sources.bib

@@ -0,0 +1,24 @@
+@misc{Ziltner2024,
+  author = {Ziltner, Prof. Dr., Fabian},
+  month = dec,
+  title = {{Notizen zur Vorlesung Analysis 1 für ITET und RW, Herbstsemester
+           2024}},
+  year = {2024},
+}
+
+@misc{Struwe2010,
+  author = {Struwe, Prof. Dr., Michael},
+  month = nov,
+  title = {{Analysis für Informatik}},
+  year = {2010},
+}
+
+@software{MCD2024,
+  author = {{The Manim Community Developers}},
+  license = {MIT},
+  month = apr,
+  title = {{Manim - Mathematical Animation Framework}},
+  url = {https://www.manim.community/},
+  version = {v0.18.1},
+  year = {2024},
+}

fs25/analysis_I_II/umkehrsatz/umkehrsatz.tex

@@ -0,0 +1 @@
+\chapter{Umkehrsatz, Satz über implizite Funktionen, Untermannigfaltigkeit des Koordinatenraums, Tangentialraum}

fs25/analysis_I_II/vektorfeld/vektorfeld.tex

@@ -0,0 +1 @@
+\chapter{Vektorfelder und die Sätze von Green, Stokes und Gauss}

fs25/analysis_I_II/zahlen_und_vektoren/zahlen_und_vektoren.tex

@@ -0,0 +1,3 @@
+\chapter{Zahlen und Vektoren}
+
+\newpage