diff --git a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf index 368ef21..1766e03 100644 Binary files a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf and b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf differ 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 ef06cc0..ff98091 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 @@ -1,8 +1,6 @@ \documentclass{article} -\newcommand{\dir}{~/projects/latex} -\input{\dir/include.tex} -\load{recommended} +\input{~/projects/latex/dist/recommended.tex} \setupCheatSheet{Analysis II} @@ -59,13 +57,16 @@ If you discover any errors, please open an issue or fix the issue yourself and t This Cheat-Sheet was designed with the HS2025 page limit of 10 A4 pages in mind. Thus, the whole Cheat-Sheet can be printed full-sized, if you exclude the title page, contents and this page. You could also print it as two A5 pages per A4 page and also print the -\color{MidnightBlue}\fbox{\href{https://github.com/janishutz/eth-summaries/blob/master/semester2/analysis-i/cheat-sheet.pdf}{Analysis I summary}}\color{black} +\hlhref{https://github.com/janishutz/eth-summaries/blob/master/semester2/analysis-i/cheat-sheet-jh/cheat-sheet-en.pdf}{Analysis I summary} \smallhspace in the same manner, allowing you to bring both to the exam. And yes, she did really miss an opportunity there with the quote\dots But she was also sick, so it's not as unexpected +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 │ % ╰────────────────────────────────────────────────╯ @@ -76,10 +77,23 @@ And yes, she did really miss an opportunity there with the quote\dots But she wa \input{parts/diffeq/linear-ode/01_order-one.tex} \input{parts/diffeq/linear-ode/02_constant-coefficient.tex} +\newsection \section{Differential Calculus in Vector Space} \input{parts/vectors/differentiation/00_continuity.tex} \input{parts/vectors/differentiation/01_partial_derivatives.tex} +\input{parts/vectors/differentiation/02_differential.tex} +\input{parts/vectors/differentiation/03_higher_diff.tex} +\input{parts/vectors/differentiation/04_change_of_variable.tex} +\input{parts/vectors/differentiation/05_taylor_polynomials.tex} +\input{parts/vectors/differentiation/06_critical_points.tex} +\newsection +\section{Integral Calculus in Vector Space} +\input{parts/vectors/integration/00_line_integrals.tex} +\input{parts/vectors/integration/01_int_in_rn.tex} +\input{parts/vectors/integration/02_improper_int.tex} +\input{parts/vectors/integration/03_change_of_variable_formula.tex} +\input{parts/vectors/integration/04_green_formula.tex} \end{document} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex index 590afb3..432061b 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex @@ -8,17 +8,36 @@ $\{ y = (y_1, \ldots, y_n) \in \R^n : |x_i - y_i| < \delta \smallhspace \forall \shortex \bi{(1)} $\emptyset$ and $\R^n$ are both open and closed. \bi{(2)} Open ball $D = \{ x \in \R^n : ||x - x_0|| < r \}$ is open in $\R^n$ ($x_0$ the center and $r$ radius) \bi{(3)} $I_1 \times \dots \times I_n$ is open in $\R^n$ for $I_i$ open -\bi{(4)} $X \subseteq \R^n$ open $\Leftrightarrow$ $\forall x \in X \exists \delta > 0$ s.t. open ball of center $x$ and radius $\delta$ is contained in $X$ - +\bi{(4)} $X \subseteq \R^n$ open $\Leftrightarrow$ $\forall x \in X \exists \delta > 0$ s.t. open ball of center $x$ and radius $\delta$ is contained in $X$\\ +% ──────────────────────────────────────────────────────────────────── \compactdef{Partial derivative} Let $X \subseteq \R^n$ open, $f: X \rightarrow \R^m$ and $1 \leq i \leq n$. Then $f$ has partial derivative on $X$ with respect to the $i$-th variable (or coordinate), if $\forall x_0 = (x_{0, 1}, \ldots, x_{0, n}) \in X$, $g(t) = f(x_{0, 1}, \ldots, x_{0, i - 1}, t, x_{0, i + 1}, x_{0, n})$ on set $I = \{ t \in \R : (x_{0, 1}, \ldots, x_{0, i - 1}, t, x_{0, i + 1}, \ldots, x_{0, n}) \in X \}$ is differentiable at $t = x_{0, i}$. The derivative $g'(x_{0, i})$ at $x_{0, i}$ is denoted: $\frac{\partial f}{\partial x_i}(x_0), \partial_{x_i} f(x_0) \text{ or } \partial_i f(x_0)$\\ -% +% ──────────────────────────────────────────────────────────────────── +\stepLabelNumber{all} \shortproposition Let $X \subseteq \R^n$ open, $f, g : X \rightarrow \R^m$ and $1 \leq i \leq n$. Then: \bi{(1)} If $f$ \& $g$ have $\partial_i$ on $X$, then so does $f + g$ and $\partial_{x_i} (f + g) = \partial_{x_i}(f) + \partial_{x_i}(g)$ \bi{(2)} If $m = 1$ (i.e. $\R^1$) and $f$ \& $g$ have $\partial_i$ on $X$, then so does $fg$ and $\partial_{x_i} (fg) = \partial_{x_i}(f)g + f \partial_{x_i}(g)$ and if $g(x) \neq 0 \smallhspace \forall x \in X$, then if $f \div g$ has $\partial_i$ on $X$, then so does $f \div g$ and -$\partial_{x_i}(f \div g) = (\partial_{x_i}(f) g - f \partial_{x_i}(g)) \div g^2$ +$\partial_{x_i}(f \div g) = (\partial_{x_i}(f) g - f \partial_{x_i}(g)) \div g^2$\\ +% ──────────────────────────────────────────────────────────────────── +\compactdef{Jacobi Matrix $J$} Element $J_ij = \partial_{x_j} f_i(x)$ for function $f: X \rightarrow \R^m$ with $X \subseteq \R^n$ open. $x_j$ is the $j$-th variable, +$f_i$ is the $i$-th component of the equation (i.e. in the vector of the function). $J$ has $m$ rows and $n$ columns.\\ +% ──────────────────────────────────────────────────────────────────── + +\drmvspace\drmvspace +\stepLabelNumber{all} +\compactdef{Gradient, Divergence} for $f : X \rightarrow \R$ with $X \in \R^n$ open, the \bi{gradient} is given by +$\nabla f(x_0) = \begin{pmatrix} + \partial_{x_1} f(x_0) \\ + \vdots \\ + \partial_{x_n} f(x_0) + \end{pmatrix}$ +and the + +\drmvspace\rmvspace +trace of the Jacobi Matrix, $\text{div}(f)(x_0) = \text{Tr}(J_f(x_0)) = \sum_{i = 1}^{n} \partial_{x_i} f_i(x_0)$ is called the \bi{divergence} of $f$ at $x_0$. +\rmvspace diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex new file mode 100644 index 0000000..ad72069 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex @@ -0,0 +1,14 @@ +\newsectionNoPB +\subsection{The differential} +\setLabelNumber{all}{2} +\compactdef{Differentiable function} We have function $f: X \rightarrow \R^m$, linear map $u : \R^n \rightarrow \R^m$ and $x_0 \in X$. $f$ is differentiable at $x_0$ with differential $u$ if +$\displaystyle \lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} \frac{1}{||x - x_0||} (f(x) - f(x_0) - u(x - x_0) = 0$ where the limit is in $\R^m$. +We denote $\dx f(x_0) = u$. +If $f$ is differentiable at every $x_0 \in X$, then $f$ is differentiable on $X$ + +% ──────────────────────────────────────────────────────────────────── +\stepLabelNumber{all} +\shortproposition Let $f, g : X \rightarrow \R^m$ with $X \subseteq \R^n$ open +\begin{itemize}[noitemsep] + \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$ +\end{itemize} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/03_higher_diff.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/03_higher_diff.tex new file mode 100644 index 0000000..0de9c69 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/03_higher_diff.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Higher derivatives} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex new file mode 100644 index 0000000..68a07b7 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Change of variable} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex new file mode 100644 index 0000000..d79479f --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Taylor polynomials} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex new file mode 100644 index 0000000..2cdb5ea --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Critical points} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex new file mode 100644 index 0000000..b344fac --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Line integrals} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex new file mode 100644 index 0000000..bd650bb --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Riemann integral in Vector Space} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex new file mode 100644 index 0000000..ad540dd --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Improper integrals} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex new file mode 100644 index 0000000..59d1059 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{Change of Variable Formula} 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 new file mode 100644 index 0000000..39afce6 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex @@ -0,0 +1,2 @@ +\newsectionNoPB +\subsection{The Green Formula} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/intro.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/intro.tex deleted file mode 100644 index e69de29..0000000