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z0zA&&C{Rw`KaVpsBfWOv2G6<6$kh29J|Bs{aClzF+tT*H)X-ILZy)o-zxb?Z?G3(;YOn$_xuc`OFcl^9R z$9=B&j3-eXMI6Q%#Tn`|LAJ{J79lMCNn%=BwA$@D4I8>@@~IXPXTf_?z3LYCrmImF zQA8G%;}E?7>A1V=_;mQ0*V|GzaVBi7xc9fVn`#!!myh2$53V`HKy9;~>iDUuIJk%-xcYomHS=g%DLj-SMesgKRx^srYtX z5!;1_wli*eT*7krWWEI8?FbRcVYzc}S>N7H>y9`_C^BaFt`*h!xH(p4*{C@I`%DQw zN#IFDxQ+20XwEfNzGfg8TmzaAJYWx5C-|XaU;OOiIylJCwaPV?~cu;6) JBo(Dl{uhFCc%1+M literal 0 HcmV?d00001 diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.tex index 57cece2..aa3c85d 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.tex @@ -8,11 +8,9 @@ \renewcommand{\chaptername}{Parte} \addto\captionsitalian{\renewcommand{\chaptername}{Parte}} - - \title{\Huge{Schede riassuntive di \\ \textit{Geometria e topologia differenziale}}} \date{A.A. 2025-2026} -\author{A cura di Gabriel Antonio Videtta\footnote{Basato su un layout di \underline{Luca Lombardo} e di \underline{Francesco Sorce}.} \\ \href{mailto:g.videtta1@studenti.unipi.it}{\texttt{g.videtta1@studenti.unipi.it}} \\[0.3in] Testo basato sul contenuto del corso del prof. Lisca \\ tenutosi presso l'Università di Pisa.} +\author{A cura di Gabriel Antonio Videtta\footnote{Basato su un layout di \underline{Luca Lombardo} e di \underline{Francesco Sorce}.} \\ \href{mailto:g.videtta1@studenti.unipi.it}{\texttt{g.videtta1@studenti.unipi.it}} \\[0.3in] Testo basato sul contenuto del corso del prof.~Lisca \\ tenutosi presso l'Università di Pisa.} \begin{document} \maketitle diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex index 164eacf..e2ceab0 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex @@ -1,26 +1,33 @@ \usepackage[top=1.5cm,bottom=1.5cm,left=1.5cm,right=1.5cm]{geometry} \usepackage[utf8]{inputenc} \usepackage[italian]{babel} -\usepackage{amsmath,amssymb,amsfonts,amsthm,stmaryrd} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{stmaryrd} \usepackage{mathrsfs} % per mathscr -\usepackage{graphicx}% ruota freccia per le azioni -\usepackage{marvosym}% per il \Lightning +\usepackage{graphicx} % ruota freccia per le azioni +\usepackage{marvosym} % per il \Lightning \usepackage{array} \usepackage{faktor} % per gli insiemi quoziente \usepackage[colorlinks=false]{hyperref} -\usepackage{xparse} % Per nuovi comandi con tanti input opzionali +\usepackage{xparse} % per nuovi comandi con tanti input opzionali \usepackage{relsize} % per \mathlarger \usepackage{tikz-cd} \usepackage{multicol} \usepackage{multirow} \usepackage{cancel} -\usepackage{fourier} \usepackage{enumerate} \usepackage{soul} \usepackage{nicefrac} \usepackage{longtable} \usepackage{pdflscape} \usepackage{mathtools} +\usepackage{lmodern} +\usepackage{accents} + +\renewcommand{\vec}[1]{{\underaccent{\bar}{{#1}}}} \newtheorem*{warn}{\warning \; Attenzione} @@ -36,8 +43,6 @@ {\topsep}{\topsep} {\normalfont}{}{\itshape}{.}{\newline}{} -\usepackage{fourier} - \theoremstyle{customth} \newtheorem{theorem}{Teorema}[chapter] \newtheorem{lemma}[theorem]{Lemma} @@ -52,100 +57,57 @@ \newtheorem{notation}[theorem]{Notazione} \newtheorem{example}[theorem]{Esempio} -\DeclareMathOperator{\BinNeg}{BinNeg} -\DeclareMathOperator{\Geom}{Geom} -\DeclareMathOperator{\Poisson}{Poisson} - \makeatletter \renewenvironment{proof}[1][\proofname]{\par \pushQED{\qed}% \normalfont \topsep6\p@\@plus6\p@\relax \trivlist \item[\hskip\labelsep - \itshape - #1\@addpunct{.}]\mbox{}\\* + \itshape + #1\@addpunct{.}]\mbox{}\\* }{% \popQED\endtrivlist\@endpefalse } \makeatother %============ Simboli standard ================= -\newcommand{\FF}{\mathcal{F}} -\newcommand{\PP}{\mathcal{P}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} -\newcommand{\pp}{\text{p\hspace{-0.7em}\raisebox{-3.4pt}{--}}\,\,} -\newcommand{\pbern}{\pp} - - \newcommand{\defeq}{\overset{\mathrm{def}}{=}} -\newcommand{\deq}{\overset{\mathrm{(d)}}{=}} -\newcommand{\toprob}{\overset{\mathbb{P}}{\to}} -\DeclareMathOperator{\VA}{VA} + \DeclareMathOperator{\im}{im} -\DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\shape}{S} -\DeclareMathOperator{\Exp}{Exp} - -\DeclareMathOperator{\CI}{CI} \newcommand{\eps}{\varepsilon} +\renewcommand{\phi}{\varphi} + +\newcommand{\der}[2]{\frac{\mathop{}\!\textnormal{\slshape d} #1}{\mathop{}\!\textnormal{\slshape d} #2}} +\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} \newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} \newcommand{\dx}{\dif{x}} \newcommand{\dy}{\dif{y}} -\newcommand{\dz}{\dif{z}} +\newcommand{\du}{\dif{u}} +\newcommand{\dv}{\dif{v}} +\newcommand{\ds}{\dif{s}} \newcommand{\dt}{\dif{t}} -\newcommand{\dP}{\dif{P}} -\newcommand{\dm}{\dif{\mathrm{\ \!\!m}}} - %\setcounter{secnumdepth}{1} -\newcommand{\groupto}{\rightrightarrows} - \newcommand{\restr}[2]{ - #1\arrowvert_{#2} + #1\arrowvert_{#2} } -\makeatletter -\def\moverlay{\mathpalette\mov@rlay} -\def\mov@rlay#1#2{\leavevmode\vtop{% - \baselineskip\z@skip \lineskiplimit-\maxdimen - \ialign{\hfil$\m@th#1##$\hfil\cr#2\crcr}}} -\newcommand{\charfusion}[3][\mathord]{ - #1{\ifx#1\mathop\vphantom{#2}\fi - \mathpalette\mov@rlay{#2\cr#3} - } - \ifx#1\mathop\expandafter\displaylimits\fi} -\makeatother - -\newcommand{\cupdot}{\charfusion[\mathbin]{\cup}{\cdot}} -\newcommand{\bigcupdot}{\charfusion[\mathop]{\bigcup}{\cdot}} - -\newcommand{\goesup}{\nearrow} -\newcommand{\goesdown}{\searrow} -\newcommand{\qc}{q.c.\ \!} -\newcommand{\qo}{q.o.\ \!} -\newcommand{\va}{v.a.\ \!} - -\newcommand{\BB}{\mathcal{B}} -\newcommand{\EE}{\mathbb{E}} -\DeclareMathOperator{\Var}{Var} -\DeclareMathOperator{\Cov}{Cov} - -\newcommand{\cS}{\mathcal{S}} - \newcommand{\inv}{^{-1}} \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} - \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{quiver}[2021/01/11 quiver] @@ -160,12 +122,12 @@ % A TikZ style for curved arrows of a fixed height, due to AndréC. \tikzset{curve/.style={settings={#1},to path={(\tikztostart) - .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) - and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) - .. (\tikztotarget)\tikztonodes}}, - settings/.code={\tikzset{quiver/.cd,#1} - \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}}, - quiver/.cd,pos/.initial=0.35,height/.initial=0} + .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + .. (\tikztotarget)\tikztonodes}}, + settings/.code={\tikzset{quiver/.cd,#1} + \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}}, + quiver/.cd,pos/.initial=0.35,height/.initial=0} % TikZ arrowhead/tail styles. \tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex index a37d85a..61194cf 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex @@ -4,10 +4,27 @@ \setlength{\parindent}{2pt} \begin{multicols*}{2} - \section*{Analisi o quel che è} - \addcontentsline{toc}{section}{Analisi o quel che è} + Impiegheremo caratteri latini ($x$, $y$, $z$) per indicare quantità reali; + caratteri latini sottolineati ($\vec{x}$, $\vec{y}$, $\vec{z}$) per indicare + vettori o campi vettoriali; caratteri greci minuscoli ($\alpha$, $\beta$, $\gamma$) per indicare + curve; e caratteri greci maiuscoli ($\Sigma$) per indicare superfici. + + \section*{Analisi matematica} + \addcontentsline{toc}{section}{Analisi matematica} \begin{itemize} - \item uwu. + \item $f_i$ -- nel caso di una funzione $f : \RR^n \to \RR^m$, la proiezione di $f$ sulla $i$-esima coordinata, ovverosia + $\pi_i \circ f : \RR^n \to \RR$. + \item $\der{f}{t}(x)$, $f'(x)$ -- derivata di una funzione $f : I \subseteq \RR \to \RR^n$. Nel caso $n > 1$, coincide con + il vettore $({f_i}'(t))_i)$. La notazione può essere iterata per ottenere le derivate successive. + \item $\partial_{x_i} f(\vec{x})$, $\pd{f}{x_i}(\vec{x})$, $f_{x_i}(\vec{x})$ -- derivata parziale nella $i$-esima coordinata + di una funzione $f : \RR^n \to \RR$ nel punto $\vec{x}$. La notazione può essere iterata per ottenere le + derivate successive. + \item $\nabla f(\vec{x})$ -- gradiente di una funzione $f : \RR^n \to \RR$, ovverosia il vettore $(\partial_{x_i} f(\vec{x}))_i^\top$. + \item $J f(\vec{x})$ -- jacobiano di una funzione $f : \RR^n \to \RR^m$ nel punto $x$, ovverosia la matrice + $(\partial_{x_j} f_i (\vec{x}))_{i, j} = (\nabla f_i(\vec{x}))_{i}$. + \item $J_{\vec{y}} f(\vec{p})$ -- nel caso di una funzione $f : \RR^{m} \times \RR^{n} \to \RR^{n}$, + la sottomatrice $n \times n$ quadrata di $J f(\vec{p})$ date dalle ultime $n$ colonne. Coincide + con $J (\pi_{\RR^n} \circ f)(\vec{p})$. \end{itemize} \end{multicols*} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex index df1c8ae..c03e209 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex @@ -5,11 +5,61 @@ \begin{multicols*}{2} - \section*{Analisi matematica e teoria della misura} - \addcontentsline{toc}{section}{Analisi matematica e teoria della misura} + \section*{Analisi matematica} + \addcontentsline{toc}{section}{Analisi matematica} \begin{itemize} - \item $\text{uwu}^2$. + \item \textbf{Teorema della funzione implicita} -- Sia $U$ un aperto di $\RR^m \times \RR^n$ e sia + $f : U \to \RR^n$ una funzione di classe $C^k$ con $k \geq 1$. Sia $\vec{p} = (\vec{x_0}, \vec{y_0})$ un punto in $U$ con + $f(\vec{x_0}, \vec{y_0}) = \vec{a}$ e $J_{\vec{y}} f(\vec{p})$ invertibile. \smallskip + + Allora esiste un + intorno $A = I_{\vec{x}} \times I_{\vec{y}}$ di $\vec{p}$ in $U$ all'interno del quale esiste un'unica + funzione $g : I_{\vec{x}} \to I_{\vec{y}}$ di classe $C^k$ per cui: + \[ \vec{y} = g(\vec{x}) \iff f(\vec{x}, \vec{y}) = \vec{a}, \quad \text{(in $A$)}. \] + Inoltre per tale $g$ vale: + \[ J g(\vec{x_0}) = - J_{\vec{y}} f(\vec{p})\inv J_{\vec{x}} f(\vec{p}). \] + + \item \textbf{Teorema di invertibilità locale} -- Sia $U$ un aperto di $\RR^n$ e sia $f : U \to \RR^n$ una + funzione di classe $C^k$, con $k \geq 1$. Sia $\vec{x_0}$ un punto in $U$ con $J f(\vec{x_0})$ invertibile. \smallskip + + Allora esiste un + intorno $A$ di $\vec{x_0}$ in $U$ dentro al quale $\restr{f}{A}$ ha un'inversa $g$, anch'essa di classe + $C^k$, per la quale $J g(f(\vec{x})) = J f(\vec{x})\inv$. + + \item \textbf{Teorema di esistenza e unicità globale per sistemi lineari} -- + Sia $I \subseteq \RR$ un intervallo aperto e siano date due funzioni continue: + \[ A : I \to \RR^{n \times n} \quad \text{e} \quad \vec{b} : I \to \RR^n. \] + Fissato $(t_0, \vec{y_0}) \in I \times \RR^n$, esiste un'unica soluzione $\vec{y} : I \to \RR^n$ + del problema di Cauchy: + \[ + \begin{cases} + \vec{y}'(t) = A(t)\vec{y}(t) + \vec{b}(t), \\ + \vec{y}(t_0) = \vec{y_0}. + \end{cases} + \] + + \item \textbf{Teorema di Cauchy-Lipschitz per l'esistenza e l'unicità locale} -- + Sia $\Omega$ un aperto di $\RR \times \RR^n$ e sia $\vec{f} : \Omega \to \RR^n$ una funzione continua. + Si supponga inoltre che $\vec{f}$ sia \emph{localmente lipschitziana} rispetto alla seconda variabile. \smallskip + + Allora, per ogni $(t_0, \vec{y_0}) \in \Omega$, esistono $\delta > 0$ e un'unica funzione + $\vec{y} : (t_0 - \delta, t_0 + \delta) \to \RR^n$ di classe $C^1$ che risolve il problema di Cauchy: + \[ + \begin{cases} + \vec{y}'(t) = \vec{f}(t, \vec{y}(t)) \\ + \vec{y}(t_0) = \vec{y_0} + \end{cases} + \] + + \item \textbf{Teorema di dipendenza liscia dai dati iniziali} -- + Sia $\Omega$ un aperto di $\RR \times \RR^n$ e sia $\vec{f} : \Omega \to \RR^n$ una funzione di classe + $C^k$ con $k \geq 1$. Indichiamo con $\Phi(t, t_0, \vec{y_0})$ la soluzione massimale del + problema di Cauchy con dato iniziale $\vec{y}(t_0) = \vec{y_0}$. \smallskip + + Allora l'insieme di definizione del flusso: + \[ \mathcal{D} = \big\{ (t, t_0, \vec{y_0}) \in \RR \times \Omega : \text{la soluzione esiste al tempo } t \big\} \] + è un aperto di $\RR \times \Omega$ e l'applicazione $\Phi : \mathcal{D} \to \RR^n$ è di classe $C^k$. \end{itemize} \end{multicols*} \ No newline at end of file