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available at \hlurl{https://github.com/janishutz/summaries} + \end{Huge} +\end{center} + +\begin{large} + This allows for quicker updating in case errors are found and you will always have access to the latest version +\end{large} + +\end{document} From 9ac94af17aa482fa580d9441d86f16f2c94c2882 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 30 Dec 2025 08:35:16 +0100 Subject: [PATCH 2/3] [Analysis] Update to new helper import scheme, continue differentiability summary --- .../analysis-ii-cheat-sheet.pdf | Bin 280596 -> 292565 bytes .../analysis-ii-cheat-sheet.tex | 22 +++++++++++--- .../01_partial_derivatives.tex | 27 +++++++++++++++--- .../differentiation/02_differential.tex | 14 +++++++++ .../differentiation/03_higher_diff.tex | 2 ++ .../differentiation/04_change_of_variable.tex | 2 ++ .../differentiation/05_taylor_polynomials.tex | 2 ++ .../differentiation/06_critical_points.tex | 2 ++ .../vectors/integration/00_line_integrals.tex | 2 ++ .../vectors/integration/01_int_in_rn.tex 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z1{kudUOuuRRq9VeN)1%UY0hU^hlL*H3yxDID=r&VQj_3PUMd$hE4rl!{qj;C)hK02 zFLnRBN8aO-&A7_t(_0392-oFssyCCGRgoLR7ji>f)OCi)E#+Y(@TE+Pc;YkC5S6GyaQ&2UE^s zG6jxk9in1-HM#Uqc69BR(RvF`VM%DJ^dV%R 0$ 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 From d385c9740dda9293f394805bfb389fb16522149f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 30 Dec 2025 11:32:54 +0100 Subject: [PATCH 3/3] [Analysis] Differential complete --- .../analysis-ii-cheat-sheet.pdf | Bin 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zch969ABZ|W7IRc=wXXji^Lr1B$tUt$66vDH?ZL@W3hLu(v|ro7S>TPf$1w8u?d1JF i;T*^&BGVN)B~Ql;1wJ?q%ug$ikXN7)6~!27)BG1w$+b8D diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex index 0e5b451..4f4e1d1 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex +++ b/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 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 index ad72069..50b9462 100644 --- 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 @@ -1,14 +1,57 @@ -\newsectionNoPB +\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 $X$ - +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 $\dx (f + g) = \dx f + \dx g$ + \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 $w_1 + w_2$