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z?VQjTKpH)}p~MzKhCRm{HjVtsJd=La-&;{dpM9f>eDg{}x$hRvPlkxsy|sE)RJ}iY?NQEnD>@8cY`RW#5`ng4 z^`Y4%O>m4fjK0%>zvjE<=D^+lz?p6*okkVz#^%u3wmfFyrcOFH{6ZgHeNKXa3Og3K zU7g=6Qu-{1(Q8k=myx2xH^=MNUw1@2j?)LDqsx& zw|PYM)E#;gSguAG;5cFAbo+j))`I{iB1oP3I+941!F{Bq8 q3Sd~iA&q?{GyeZ-E4#QFJG**1n_Hl8ak6u 0$ 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: - \[ - \mathcal{D} = \left\{ (t, t_0, \vec{y_0}) \in \RR \times \Omega : - \begin{array}{c} - \text{la soluzione } \vec{y}(t, t_0, \vec{y_0}) \\ - \text{esiste al tempo } t - \end{array} - \right\} - \] - è un aperto di $\RR \times \Omega$ e l'applicazione $\Phi : \mathcal{D} \to \RR^n$ è di classe $C^k$. - - \item \textbf{Caratterizzazione dell'annullamento della misura di Lebesgue} -- Sia $A$ un sottinsieme di $\RR^n$. Allora - $A$ ha misura nulla se e solo se, per ogni scelta di $\eps > 0$, esiste una famiglia numerabile $\{B_i\}_{i \geq 0}$ - di rettangoli $B_i \subseteq \RR^n$ tali per cui: - \[ A \subseteq \bigcup_{i \geq 0} B_i, \quad \sum_{i \geq 0} \vol(B_i) < \eps. \] - - \item \textbf{Lemma per la nullità della misura su un unione numerabile di insiemi di misura nulla} -- Se $\{A_k\}_{k \geq 0}$ - è una famiglia di sottinsiemi di misura nulla di $\RR^n$, allora anche $\bigcup_{k \geq 0} A_k$ ha misura nulla. - - \item \textbf{Teorema di Sard sugli aperti di $\RR^n$} -- Sia $U \subseteq \RR^n$ aperto, e sia $f : U \to \RR^n$ una mappa - liscia. Allora l'insieme $\crit(f)$ dei valori critici di $f$ ha misura nulla. - \end{itemize} + \section*{Analisi matematica e teoria della misura} + \addcontentsline{toc}{section}{Analisi matematica e teoria della misura} + + \begin{itemize} + \item \textbf{Teorema di Schwarz} -- Sia $f : \RR^n \to \RR$ una funzione che ammette derivate seconde + miste continue in $\vec{x}$. Allora $\partial_{x_i x_j} f(\vec{x}) = \partial_{x_j x_i} f(\vec{x})$ per + ogni variabile $x_i$, $x_j$. + + \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} (o \textit{della funzione inversa}) -- 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 di equazioni differenziali} -- + 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: + \[ + \mathcal{D} = \left\{ (t, t_0, \vec{y_0}) \in \RR \times \Omega : + \begin{array}{c} + \text{la soluzione } \vec{y}(t, t_0, \vec{y_0}) \\ + \text{esiste al tempo } t + \end{array} + \right\} + \] + è un aperto di $\RR \times \Omega$ e l'applicazione $\Phi : \mathcal{D} \to \RR^n$ è di classe $C^k$. + + \item \textbf{Caratterizzazione dell'annullamento della misura di Lebesgue} -- Sia $A$ un sottinsieme di $\RR^n$. Allora + $A$ ha misura nulla se e solo se, per ogni scelta di $\eps > 0$, esiste una famiglia numerabile $\{B_i\}_{i \geq 0}$ + di rettangoli $B_i \subseteq \RR^n$ tali per cui: + \[ A \subseteq \bigcup_{i \geq 0} B_i, \quad \sum_{i \geq 0} \vol(B_i) < \eps. \] + + \item \textbf{Lemma per la nullità della misura su un unione numerabile di insiemi di misura nulla} -- Se $\{A_k\}_{k \geq 0}$ + è una famiglia di sottinsiemi di misura nulla di $\RR^n$, allora anche $\bigcup_{k \geq 0} A_k$ ha misura nulla. + + \item \textbf{Teorema di Sard sugli aperti di $\RR^n$} -- Sia $U \subseteq \RR^n$ aperto, e sia $f : U \to \RR^n$ una mappa + liscia. Allora l'insieme $\crit(f)$ dei valori critici di $f$ ha misura nulla. + \end{itemize} \end{multicols*} \ No newline at end of file diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex index ab4912f..6f0b6c1 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex @@ -64,7 +64,7 @@ si estende all'identità. \end{proof} - \subsection{Varietà differenziabili, carte, atlanti e parametrizzazioni locali} + \subsection{Varietà differenziabili, carte, atlanti, parametrizzazioni locali e funzioni di transizione} \begin{definition}[Varietà differenziabile liscia senza bordo] Un insieme $M \subseteq \RR^k$ si dice \textbf{varietà (differenziabile liscia senza bordo) di dimensione $m>0$} (o $m$-varietà) se per ogni @@ -93,6 +93,13 @@ regolari. \end{remark} + \begin{definition}[Funzione di transizione] + Date due parametrizzazioni locali $f : U \to f(U)$ e $g : V \to g(V)$ con intersezione + delle immagini \underline{non} vuota, si + definisce la \textbf{funzione di transizione da $f$ a $g$} come la seguente funzione: + \[ \boxed{g\inv \circ f : f\inv(g(V)) \to g\inv(f(U)).} \] + \end{definition} + \subsection{Prodotto di varietà} \begin{proposition}[Prodotto di varietà] \label{prop:prodotto_varietà} @@ -288,7 +295,7 @@ Sia $f : M \to N \times O$ una mappa liscia, dove $M$, $N$ e $O$ sono varietà. Se $f(x) = (g(x), p(x))$, allora $g$ e $p$ sono lisce e vale: \[ - \boxed{df_x(h) = (dg_x(h), dp_x(h)).} + \boxed{\dif f_x(h) = (\dif g_x(h), \dif p_x(h)).} \] \end{proposition} @@ -493,11 +500,11 @@ \] Allora, per la Proposizione \ref{prop:diff_prodotto}, vale: \[ - dF_x(v) = (df_x(v), dL_x(v)) = (df_x(v), L(v)), + \dif F_x(v) = (\dif f_x(v), \dif L_x(v)) = (\dif f_x(v), L(v)), \] - dove si è usato che $L$ è una mappa lineare. $dF_x(v)$ si annulla solo + dove si è usato che $L$ è una mappa lineare. $\dif F_x(v)$ si annulla solo per $v = 0$, essendo $\restr{L}{T_x M}$ un isomorfismo; quindi - $dF_x(v)$ è invertibile, e $x$ è regolare per $F$. \smallskip + $\dif F_x(v)$ è invertibile, e $x$ è regolare per $F$. \smallskip Osserviamo che $F$ è una mappa tra varietà della stessa dimensione (vd. Proposizione \ref{prop:prodotto_varietà}), e quindi, per la Proposizione @@ -590,12 +597,20 @@ \] \end{definition} + \begin{remark} \label{rmk:hn_diffeo_rn-1} + Osserviamo che in modo naturale vale il seguente diffeomorfismo: + \[ + \boxed{\partial H^n = \{ x \in \RR^n \mid x_n = 0 \} \cong \RR^{n-1}.} + \] + \end{remark} + \begin{definition}[$m$-varietà con bordo] Si dice che $M \subseteq \RR^k$ è una \textbf{$m$-varietà con bordo} se ogni punto di $M$ ammette un intorno diffeomorfo ad un aperto del semispazio superiore $H^n$. Gli intorni e i diffeomorfismi citati formano le \textbf{carte locali} della varietà, e le inverse di tali diffeomorfismi - sono dette \textbf{parametrizzazioni locali}. \smallskip + sono dette \textbf{parametrizzazioni locali}. Analogamente si definiscono + le \textbf{funzioni di transizione}. \smallskip Si dice \textbf{bordo} della varietà $M$ l'insieme dei punti che è immagine di un punto di $\partial H^n$ tramite qualche parametrizzazione locale, e @@ -610,6 +625,59 @@ Utilizzeremo dunque indistintamente le due caratterizzazioni. \end{remark} + \subsection{Proprietà del bordo di una varietà con bordo} + + \begin{lemma}[I punti di bordo sono sempre immagini di elementi di bordo] \label{lem:punti_di_bordo} + Sia $x$ un punto del bordo $\partial M$ di una $m$-varietà con bordo $M$. Se + $g$ è una parametrizzazione locale di $x$, allora $x$ è immagine di un + punto di bordo di $H^n$ tramite $g$. Equivalentemente, $x$ è un punto di + $\partial M$ se e solo se è immagine di un valore di bordo per + ogni sua parametrizzazione locale. + \end{lemma} + + \begin{proof} + Sia $g : U \subseteq H^m \to g(U)$ una parametrizzazione locale di $x$ con + $g(u) = x$. Poiché $x$ è un punto di $\partial M$, + allora esiste una parametrizzazione locale $f : V \subseteq H^m \to f(V)$ + di $x$ tale per cui esiste $v \in \partial V = \partial H^m \cap V$ con + $f(v) = x$. \smallskip + + Se $f = g$, la tesi è dimostrata. Se $f \neq g$ e per assurdo $u \notin \partial U$, + allora la funzione di transizione $g\inv \circ f$ si restringerebbe a + un diffeomorfismo tra un aperto di $\RR^m$ diffeomorfo a $\RR^m$ e + un aperto di $H^m$ diffeomorfo a $H^m$. Tuttavia $\RR^m$ e + $H^m$ non sono diffeomorfi, \Lightning. Dunque $u \in \partial U$. + \end{proof} + + \begin{corollary}[Il bordo si trasporta naturalmente tramite parametrizzazione locale] \label{cor:bordo_param_locale} + Sia $g : U \to g(U)$ una parametrizzazione locale di una $m$-varietà con bordo $M$. Allora: + \[ + \boxed{g(\partial U) = g(U) \cap \partial M.} + \] + \end{corollary} + + \begin{proof} + L'inclusione $g(\partial U) \subseteq g(U) \cap \partial M$ è ovvia. + L'inclusione opposta invece è data dal Lemma \ref{lem:punti_di_bordo}. + \end{proof} + + \begin{proposition} + Sia $M$ una $m$-varietà con bordo. Allora $\partial M$ è + una varietà senza bordo di dimensione $m-1$. + \end{proposition} + + \begin{proof} + Sia $x \in \partial M$. Allora esiste una parametrizzazione locale + $g : U \subseteq H^m \to M$ con $g(u) = x$ e $u \in \partial U = U \cap \partial H^m$. \smallskip + + La restrizione $\restr{g}{\partial U}$ è un diffeomorfismo (vd. Proposizione \ref{prop:comp_liscia}). Ricordiamo che + $g(\partial U) = g(U) \cap \partial M$ dal Corollario \ref{cor:bordo_param_locale}. Allora, poiché + $\partial H^m \cong \RR^{m-1}$, possiamo identificare + $\partial U$ come aperto in $\RR^{m-1}$. Quindi + $(\restr{g}{\partial U})\inv$ induce una carta locale per $\partial M$, + e si conclude che $\partial M$ è una varietà di dimensione $m-1$. \smallskip + \end{proof} + \subsection{Differenziale e spazio tangente su varietà con bordo} \begin{remark}[Il differenziale sul bordo di $H^n$ è ben definito] @@ -633,15 +701,15 @@ \begin{definition}[Differenziale su $H^n$] Sia $g : U \to \RR^k$ una mappa liscia da un aperto $U \subseteq H^n$. \smallskip - Per $x \in U \setminus \partial H^n$, il differenziale $dg_x$ è + Per $x \in U \setminus \partial H^n$, il differenziale $\dif g_x$ è definito come l'usuale differenziale dato dalla restrizione di $g$ a un aperto di $\RR^n$. \smallskip - Per $x \in U \cap \partial H^n$, il differenziale $dg_x$ è indotto dal + Per $x \in U \cap \partial H^n$, il differenziale $\dif g_x$ è indotto dal differenziale di una qualsiasi estensione $\tilde{g}$ di $g$ in un intorno aperto di $x$, ovverosia: \[ - \boxed{dg_x \defeq d \hat{g}_x.} + \boxed{\dif g_x \defeq \dif \hat{g}_x.} \] \end{definition} @@ -674,48 +742,6 @@ concetti di punto regolare/critico e di valore regolare/critico. \end{remark} - \subsection{Bordo di una varietà} - - \begin{proposition} - Sia $M$ una $m$-varietà con bordo. Allora $\partial M$ è - una varietà senza bordo di dimensione $m-1$. - \end{proposition} - - \begin{remark} - Osserviamo innanzitutto che $\partial H^m \cong \RR^{m-1}$, - su cui si fonda l'idea della dimostrazione. - \end{remark} - - \begin{proof} - Sia $x \in \partial M$. Allora esiste una parametrizzazione locale - $g : U \subseteq H^m \to M$ con $g(u) = x$ e $u \in \partial U = U \cap \partial H^m$. \smallskip - - La restrizione $\restr{g}{\partial U}$ è un diffeomorfismo. Mostriamo - che $g(\partial U) = g(U) \cap \partial M$: questo ci permetterebbe di concludere - che $(\restr{g}{\partial U})\inv$ è una carta locale per $\partial M$, - e quindi, poiché $\partial U \cong \RR^{m-1}$, che $\partial M$ è una varietà di dimensione $m-1$. \smallskip - - L'inclusione $g(\partial U) \subseteq g(U) \cap \partial M$ è chiara per - costruzione. \smallskip - - Supponiamo che esista $y \in g(U) \cap \partial M$ non - appartenente a $g(\partial U)$. - - \begin{itemize} - \item Dal momento che $y \in \partial M$, esiste - una parametrizzazione locale $h : V \subseteq H^m \to M$ tale per cui esiste - $v \in \partial V$ con $h(v) = y$. - \item Dacché $y \in g(U)$, ma $y \notin g(\partial U)$, esiste $u' \notin \partial U$ - tale per cui $g(u') = y$. - \end{itemize} - - A meno di restringimenti di $g$, consideriamo la funzione di transizione - $h\inv \circ g$. Dacché è composizione di diffeomorfismi, $h\inv \circ g$ stessa - è un diffeomorfismo liscio. Dal momento che $u' \notin \partial U$, - $h\inv \circ g$ induce un diffeomorfismo tra un intorno aperto di $u'$ in $\underline{\RR^m}$ e uno - di $v$; tuttavia questo è assurdo perché implicherebbe $H^m \cong \RR^m$. - \end{proof} - \subsection{Varietà con bordo da valori regolari} \begin{lemma}