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MAz^1%)KVh(FTs5HOaK4? diff --git a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex index 861f706..af5f916 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.tex @@ -50,6 +50,7 @@ \newpage +\setcounter{section}{-1} \section{Introduction} 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. @@ -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} -% TODO: Everywhere: Check with TA notes to add tips and tricks - % ╭────────────────────────────────────────────────╮ % │ Content │ % ╰────────────────────────────────────────────────╯ \newsection +\input{parts/00_intro.tex} + \section{Differential Equations} \input{parts/diffeq/00_intro.tex} \input{parts/diffeq/linear-ode/00_intro.tex} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/00_intro.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/00_intro.tex new file mode 100644 index 0000000..50f3816 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/00_intro.tex @@ -0,0 +1,3 @@ +\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. diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex index 9b9263b..152c79e 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex @@ -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: \drmvspace \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*} \stepLabelNumber{all}\dhrmvspace @@ -23,7 +23,7 @@ $\gamma_i$ as above, then \end{align*} \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. We can use Green's Formula to compute integrals. For that we need the set of curves that define the set.