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\newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} + \newcommand{\dx}{\dif{x}} \newcommand{\dy}{\dif{y}} \newcommand{\du}{\dif{u}} 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 61194cf..3a71d25 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 @@ -4,27 +4,30 @@ \setlength{\parindent}{2pt} \begin{multicols*}{2} - 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. + 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} + \section*{Analisi matematica} + \addcontentsline{toc}{section}{Analisi matematica} - \begin{itemize} - \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} + \begin{itemize} + \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 $J (\pi_{\RR^n} \circ f)(\vec{p})$. + \item $C^n$ -- classe delle funzioni dotate di derivate parziali continue fino all'ordine $n$. Per $n = 0$, coincide con la classe delle funzioni continue ($C^0$). + \item $C^\infty$, liscio -- classe delle funzioni derivabili parzialmente per un numero arbitrario di volte con continuità. + \item diffeomorfismo di classe $C^k$ -- funzione di classe $C^k$ con inversa di classe $C^k$. + \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 c03e209..cebfe01 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 @@ -47,8 +47,8 @@ $\vec{y} : (t_0 - \delta, t_0 + \delta) \to \RR^n$ di classe $C^1$ che risolve il problema di Cauchy: \[ \begin{cases} - \vec{y}'(t) = \vec{f}(t, \vec{y}(t)) \\ - \vec{y}(t_0) = \vec{y_0} + \vec{y}'(t) = \vec{f}(t, \vec{y}(t)), \\ + \vec{y}(t_0) = \vec{y_0}. \end{cases} \] @@ -58,7 +58,14 @@ problema di Cauchy con dato iniziale $\vec{y}(t_0) = \vec{y_0}$. \smallskip Allora l'insieme di definizione del flusso: - \[ \mathcal{D} = \big\{ (t, t_0, \vec{y_0}) \in \RR \times \Omega : \text{la soluzione esiste al tempo } t \big\} \] + \[ + \mathcal{D} = \left\{ (t, t_0, \vec{y_0}) \in \RR \times \Omega : + \begin{array}{c} + \text{la soluzione } \vec{y}(t, t_0, \vec{y_0}) \\ + \text{esiste al tempo } t + \end{array} + \right\} + \] è un aperto di $\RR \times \Omega$ e l'applicazione $\Phi : \mathcal{D} \to \RR^n$ è di classe $C^k$. \end{itemize} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex index 630c2aa..4f2c22e 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex @@ -4,4 +4,165 @@ \begin{multicols*}{2} + Qualora non specificato, assumeremo l'utilizzo di funzioni di classe $C^\infty$. \smallskip + + Se $\vec{x} \in \RR^3$ e $f$ è una funzione relativa a una curva $\alpha$, ammettiamo l'abuso di notazione + $f(\vec{x})$, intendendo $f(\alpha\inv(\vec{x}))$ (e.g., $k(P)$ per intendere $k(\alpha\inv(P)))$). + + \section{Definizioni preliminari} + + \subsection{Curve, tracce e velocità} + + \begin{definition}[Curva parametrizzata] + Una \textbf{curva parametrizzata} (o semplicemente \textit{curva}) + è una mappa $\alpha : I \subseteq \RR \to \RR^3$ + di classe $C^\infty$, dove $I$ è un intervallo + \end{definition} + + \begin{definition}[Traccia di una curva] + Si dice \textbf{traccia} (o \textit{supporto}) di una curva + parametrizzata $\alpha : I \to \RR^3$, la sua immagine $\alpha(I)$. + \end{definition} + + \begin{definition}[Velocità di una curva] + Si definisce la \textbf{velocità} di una curva parametrizzata + $\alpha(t) = (x(t), y(t), z(t))$ come la curva indotta dalla derivata di $\alpha$: + \[ \alpha'(t) = \lim_{h \to 0} \frac{\alpha(t + h) - \alpha(t)}{h} = (x'(t), y'(t), z'(t)). \] + \end{definition} + + \subsection{Lunghezza e intuizione geometrica} + + \begin{definition}[Lunghezza di una curva] + Si definisce la \textbf{lunghezza} $\ell(\alpha)$ di una curva + $\alpha : I \to \RR^3$ come: + \[ + \ell(\alpha) \defeq \int_I \norm{\alpha'(t)} \dt. + \] + \end{definition} + + \begin{remark} + La definizione data per la lunghezza di una curva corrisponde alla nostra + idea intuitiva di lunghezza tramite i seguenti due risultati: + + \begin{enumerate} + \item \textbf{Validità sul segmento:} Su un segmento lineare $\alpha(t) = A + t(B-A)$ con $I = [0, 1]$, + $\ell(\alpha) = \norm{B-A}$. + \item \textbf{Approssimazione poligonale:} La lunghezza $\ell(\alpha)$ è il limite delle lunghezze delle poligonali + inscritte nella curva. In termini teorici: + + \begin{quote} + Sia dato $\eps > 0$. Allora esiste $\delta > 0$ tale per cui, per ogni partizione $\{t_i\}_{i=0}^n$ + di $I$ di finezza inferiore a $\delta$ (i.e., $\max \abs{t_{i+1} - t_i} < \delta$), vale $\norm{S - \ell(\alpha)} < \eps$, + dove $S \defeq \sum_{i=0}^{n-1} \norm{\alpha(t_{i+1}) - \alpha(t_{i})}$. + \end{quote} + \end{enumerate} + \end{remark} + + \section{(Ri)parametrizzazioni, regolarità e parametrizzazioni p.l.a.} + + \subsection{Riparametrizzazione e prime proprietà} + + \begin{definition}[Riparametrizzazione di una curva] + Data una curva $\alpha : I \to \RR^3$, una \textbf{riparametrizzazione $\beta$ di $\alpha$} + è una curva $\beta : J \to \RR^3$ tale per cui esiste un diffeomorfismo liscio + $h : I \to J$ con $\alpha = \beta \circ h$. + \[\begin{tikzcd} + I && {\RR^3} \\ + \\ + J + \arrow["\alpha"', from=1-1, to=1-3] + \arrow["h", from=1-1, to=3-1] + \arrow["\beta", from=3-1, to=1-3] + \end{tikzcd}\] + Se $h' > 0$, si dice che $h$ mantiene l'orientazione di $\alpha$; se $h' < 0$, + $h$ inverte l'orientazione. + \end{definition} + + \begin{proposition} + Se $\beta$ è una riparametrizzazione di $\alpha$, allora $\ell(\beta) = \ell(\alpha)$. + \end{proposition} + + \subsection{Regolarità e coordinate date dalla lunghezza d'arco} + + \begin{definition}[Curva regolare] + Si dice che una curva $\alpha : I \to \RR^3$ è \textbf{regolare} se + $\alpha'(t) \neq 0$ per ogni $t \in I$. + \end{definition} + + \begin{definition}[Curva parametrizzata a lunghezza d'arco] + Si dice che una curva $\alpha : I \to \RR^3$ è \textbf{parametrizzata + a lunghezza d'arco} (\textbf{p.l.a.}) se $\alpha'$ è un vettore + unitario (i.e., $\norm{\alpha'} = 1$). + In tal caso, $\ell\left(\restr{\alpha}{[a, b]}\right) = b-a$. + \end{definition} + + \begin{proposition}[Riparametrizzazione a lunghezza d'arco] + Se $\alpha : [a, b] \to \RR^3$ è una curva regolare, allora $\alpha$ + ammette una riparametrizzazione a lunghezza d'arco, ossia ammette + una riparametrizzazione $\beta : J \to \RR^3$ tale per cui + $\beta$ sia p.l.a. + \end{proposition} + + \begin{proof} + Poiché $\alpha$ è regolare, la funzione $s : I \to [0, \ell(\alpha)]$ tale per cui + \[ s(t) = \int_{a}^t \norm{\alpha'(t)} \dt \] + è un diffeomorfismo liscio. Quindi $\beta = \alpha \circ s\inv$ è una riparametrizzazione + di $\alpha$, e vale: + \[ + \beta'(s) = \frac{\alpha'(s\inv(s))}{s'(s\inv(s))} = \frac{\alpha'(s\inv(s))}{\norm{\alpha'(s\inv(s))}}, + \] + che è un vettore unitario. + \end{proof} + + \section{Il triedro di Frenet (caso p.l.a.)} + + In tutta questa sezione consideriamo una curva p.l.a. $\beta$. + + \subsection{Versore tangente e curvatura di una curva} + + \begin{definition}[Versore tangente] + Sia $\beta$ una curva p.l.a., allora si definisce il + suo \textbf{versore tangente} $T_\beta$ come $\beta'$. + \end{definition} + + \begin{definition}[Curvatura] + Sia $\beta$ una curva p.l.a., allora si definisce la + \textbf{curvatura} $k_\beta(s)$ di $\beta$ al tempo $s$ come + $\norm{T_\beta'(s)}$. \smallskip + + Laddove è chiaro dal contesto quale sia $\beta$, scriviamo solo $k(s)$. + \end{definition} + + \subsection{Curve di Frenet, versore normale e binormale} + + \begin{definition}[Curva di Frenet] + Una curva p.l.a.~$\beta$ si dice \textbf{curva di Frenet} se + ad ogni tempo $s$, la curvatura è positiva ($k_\beta(s) > 0$). + \end{definition} + + \begin{definition}[Versore normale] + Se $\beta$ è una curva di Frenet, allora è ben definito + a ogni tempo $s$ il \textbf{versore normale} $N_\beta(s)$ così + definito: + \[ + N_\beta(s) = \frac{T_\beta(s)}{\norm{T_\beta(s)}}. + \] + \end{definition} + + \begin{definition}[Versore binormale] + Se $\beta$ è una curva di Frenet, allora è ben definito + a ogni tempo $s$ il \textbf{versore binormale} $B_\beta(s)$ così + definito: + \[ + B_\beta(s) = T_\beta(s) \times N_\beta(s). + \] + \end{definition} + + \begin{remark}[Triedro di Frenet] + Se $\beta$ è di Frenet, allora, dacché $\dot{T_\beta} \perp T_\beta$, + $N_\beta$ e $T_\beta$ sono linearmente indipendenti. Dunque + $\{T_\beta, N_\beta, B_\beta\}$ formano una base ortonormale a ogni tempo + $s$. Tale base è detta \textbf{triedro di Frenet}. + \end{remark} + \subsection{Equazioni di Frenet} \end{multicols*}