Merge branch 'main' of https://github.com/janishutz/eth-summaries

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}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex

@@ -66,3 +66,4 @@ Furthermore: $\{ x\in \R^3 : ||x - x_0|| = r \}$ is closed\\
 \shorttheorem Let $(X \neq \emptyset) \subseteq \R^n$ compact and $f: X \rightarrow \R$ continuous. 
 Then $f$ bounded, has $\max$ and $\min$, i.e. $\exists x_+, x_- \in X$ s.t. $\displaystyle f(x_+) = \sup_{x \in X} f(x)$ and $\displaystyle f(x_-) = \inf_{x \in X} f(x)$
 % ────────────────────────────────────────────────────────────────────
+\rmvspace

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

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex

@@ -0,0 +1,57 @@
+\newsection
+\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
+\rmvspace Let $f: X \rightarrow \R^m$ be differentiable on $X$
+\begin{itemize}[noitemsep]
+    \item $f$ is continuous on $X$
+    \item $f$ admits partial derivatives on $X$ with respect to each variable
+    \item Assume $m = 1$, let $x_0 \in X$ and let $u(x_1, \ldots, x_n) = a_1 x_1 + \ldots + a_n x_n$ be diff. of $f$ at $x_0$.
+          Then $\partial_{x_i} f(x_0) = a_i$ for $1 \leq i \leq n$
+\end{itemize}
+\rmvspace
+% ────────────────────────────────────────────────────────────────────
+\stepLabelNumber{all}
+\shortproposition Let $f, g : X \rightarrow \R^m$ with $X \subseteq \R^n$ open
+\rmvspace
+\begin{itemize}[noitemsep]
+    \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$. If $m = 1$, then $fg$ is differentiable
+    \item If $m = 1$ and if $g(x) \neq 0 \forall x \in X$, then $f \div g$ is differentiable
+\end{itemize}
+\rmvspace
+% ────────────────────────────────────────────────────────────────────
+\shortproposition If $f$ as above has all partial derivatives on $X$ and if they are all continuous on $X$, then $f$ is differentiable on $X$.
+The differential is the Jacobi Matrix of $f$ at $x_0$.
+This implies that most elementary functions are differentiable.\\
+% ────────────────────────────────────────────────────────────────────
+\compactproposition{Chain Rule} For $X \subseteq \R^n$ and $Y \subseteq \R^m$ both open and $f: X \rightarrow Y$ and $g : Y \rightarrow \R^p$ are both differentiable.
+Then $g \circ f$ is differentiable on $X$ and for any $x \in X$, its differential is given by
+$\dx (g \circ f)(x_0) = \dx g(f(x_0)) \circ \dx f(x_0)$.
+The Jacobi matrix is $J_{g \circ f}(x_0) = J_g(f(x_0)) J_f(x_0)$ (RHS is a matrix product)\\
+% ────────────────────────────────────────────────────────────────────
+\setLabelNumber{all}{11}
+\compactdef{Tangent space} The graph of the affine linear approximation $g(x) = f(x_0) + u(x - x_0)$, or the set
+\vspace{-0.75pc}
+\begin{align*}
+    \{ (x, y) \in \R^n \times \R^m : y = f(x_0) + u(x - x_0) \}
+\end{align*}
+
+\drmvspace\rmvspace
+% ────────────────────────────────────────────────────────────────────
+\stepLabelNumber{all}
+\compactdef{Directional derivative} $f$ has a directional derivative $w \in \R^m$ in the direction of $v \in \R^n$,
+if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$
+% ────────────────────────────────────────────────────────────────────
+\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$.
+% ────────────────────────────────────────────────────────────────────
+\shortproposition Let $f$ as previously be differentiable. Then for any $x \in X$ and non-zero $v \in \R^n$,
+$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$
+% ────────────────────────────────────────────────────────────────────
+\shortremark The values of the above directional derivative are linear with respect to the vector $v$.
+Suppose we know the dir. der. $w_1$ and $w_2$ in directions $v_1$ and $v_2$, then the directional derivative in direction $v_1 + v_2$ is $w_1 + w_2$

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/03_higher_diff.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Higher derivatives}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Change of variable}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Taylor polynomials}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Critical points}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Line integrals}

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}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Improper integrals}

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}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{The Green Formula}