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[Analysis] Add general notes

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Janis Hutz
2026-02-01 09:59:14 +01:00
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semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf
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semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex
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@@ -50,6 +50,7 @@
\newpage \newpage
\setcounter{section}{-1}
\section{Introduction} \section{Introduction}
This Cheat-Sheet does not serve as a replacement for solving exercises and getting familiar with the content. This Cheat-Sheet does not serve as a replacement for solving exercises and getting familiar with the content.
There is no guarantee that the content is 100\% accurate, so use at your own risk. There is no guarantee that the content is 100\% accurate, so use at your own risk.
@@ -68,12 +69,12 @@ And yes, she did really miss an opportunity there with the quote\dots But she wa
This summary also uses tips and tricks from this \hlhref{https://polybox.ethz.ch/index.php/s/WBGFTRdEjRwJjQC}{Exercise Session} This summary also uses tips and tricks from this \hlhref{https://polybox.ethz.ch/index.php/s/WBGFTRdEjRwJjQC}{Exercise Session}
% TODO: Everywhere: Check with TA notes to add tips and tricks
% ╭────────────────────────────────────────────────╮ % ╭────────────────────────────────────────────────╮
% │ Content │ % │ Content │
% ╰────────────────────────────────────────────────╯ % ╰────────────────────────────────────────────────╯
\newsection \newsection
\input{parts/00_intro.tex}
\section{Differential Equations} \section{Differential Equations}
\input{parts/diffeq/00_intro.tex} \input{parts/diffeq/00_intro.tex}
\input{parts/diffeq/linear-ode/00_intro.tex} \input{parts/diffeq/linear-ode/00_intro.tex}

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semester3/analysis-ii/cheat-sheet-jh/parts/00_intro.tex Normal file
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\section{General tips}
Use systems of equations if given some points, or other optimization techniques.
The Analysis I cheat sheet has a derivatives and anti-derivatives table.

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semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex
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@@ -11,7 +11,7 @@ with $\gamma_i = (\gamma_{i, 1}, \gamma_{i, 2}) : [a_i, b_i] \rightarrow \R^2$ a
$f = (f_1, f_2)$ is a vector field of class $C^1$ on open set containing $X$. Then: $f = (f_1, f_2)$ is a vector field of class $C^1$ on open set containing $X$. Then:
\drmvspace \drmvspace
\begin{align*} \begin{align*}
\int_{X} \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial_y} \right) \dx x \dx y = \sum_{i = 1}^{k} \int_{\gamma_i} f \cdot \dx \vec{s} \int_{X} \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} \right) \dx x \dx y = \sum_{i = 1}^{k} \int_{\gamma_i} f \cdot \dx \vec{s}
\end{align*} \end{align*}
\stepLabelNumber{all}\dhrmvspace \stepLabelNumber{all}\dhrmvspace
@@ -23,7 +23,7 @@ $\gamma_i$ as above, then
\end{align*} \end{align*}
\drmvspace \drmvspace
\shade{gray}{Understanding and applying Green's Formula} The $\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{y} = \text{curl}(f)$, i.e. it is the 2D-curl of $f$. \shade{gray}{Understanding and applying Green's Formula} The $\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} = \text{curl}(f)$, i.e. it is the 2D-curl of $f$.
Thus, the sum of all line integrals is the same thing as the Riemann-Integral of the curl. Thus, the sum of all line integrals is the same thing as the Riemann-Integral of the curl.
We can use Green's Formula to compute integrals. For that we need the set of curves that define the set. We can use Green's Formula to compute integrals. For that we need the set of curves that define the set.