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z8zP=$OFqY0dGl5F8|XjsNwz{GnZ_YX0DCx9VY7eqM{HN^60McYkB^dXgk`L=-VBo@ zaTS`&d>{J!5BSsJ(4xwvtyB&U#tQkU`pV6gU%>7Yc`HfGmN$Va<-Q;*T$g821sAbA zDQgF_SnxII#rm(m{mSf5qBBty#>i20;G-Q^Id}RO3d<>hRv)5t&Mq!=lp832p7DT_a4>X*XHrnE9Q7arliL?#1b;=Nw^NKWIDs!` z9a({Cv1+Hvw`66;8R>(GwD1Su^Sd=PU()bk2uz3jEgg-VEZyK#lOGR?0v4Iq(IqnV za^aXxexkw9nW=VCNtgAKZ<-P~NtjB5<*@zAScY6RSn^SYO&i)}X z=UT=Cn%Fw0VZo8#*CPoN>{MLCrJ#ynR^(WNB?BcDbxf@oRCW3j z?p735htn)K@y|3+J=rKXLWQbJqWEC|#l@#yR zMSDZ_Crw|uk=S$+HXcLU=3n-iliajcaz9jRZ9Vu7>EyF)zpSa9@qWX0X-2KHq3FLv zWC^@1WycXgq-_C6er5Uof0P~fke|67WD$uBNAAgZ965BfYRy&D4&2`=?$D=x!@S$>&b=o0s z*1mOL 1$, 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 $\vec{x_i}$ -- derivate parziali nella $i$-esima coordinata di un campo $\vec{x}$ da $\RR^n$ a $\RR^m$, ovverosia - $((x_1)_i, \ldots, (x_m)_i)^\top$. - \item $\nabla f(\vec{x})$, $\nabla f_{\vec{x}}$ -- gradiente di una funzione $f : \RR^n \to \RR$, ovverosia il vettore $(\partial_{x_i} f(\vec{x}))_i^\top$. - \item $J f(\vec{x})$, $J f_\vec{x}$ -- 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} + \begin{itemize} + \item $\{f \operatorname{op} a\}$ -- data una funzione $f : X \to \RR$ a valori reali, $\{f \operatorname{op} a\}$ denota + l'insieme $\{ x \in X \mid f(x) \operatorname{op} a\}$, dove $a \in \RR$. + \item $f \times g$ -- date due funzioni $f : A \to C$, $g : B \to D$, la funzione $f \times g : (A \times B) \to (C \times D)$ è + definita in modo tale che $(f \times g)(a, b) = (f(a), g(b))$. + \item $\restr{f}{A}$ -- restrizione di una funzione $f$ al sottinsieme $A$ del dominio. + \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 $\der{f}{t}(x)$, $f'(x)$, $\dot{f}$ -- 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 $\vec{x_i}$ -- derivate parziali nella $i$-esima coordinata di un campo $\vec{x}$ da $\RR^n$ a $\RR^m$, ovverosia + $((x_1)_i, \ldots, (x_m)_i)^\top$. + \item $\nabla f(\vec{x})$, $\nabla f_{\vec{x}}$ -- gradiente di una funzione $f : \RR^n \to \RR$, ovverosia il vettore $(\partial_{x_i} f(\vec{x}))_i^\top$. + \item $J f(\vec{x})$, $J f_\vec{x}$ -- 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} - \section*{Geometria differenziale delle curve e delle superfici} - \addcontentsline{toc}{section}{Geometria differenziale delle curve e delle superfici} + \section*{Geometria differenziale delle curve e delle superfici} + \addcontentsline{toc}{section}{Geometria differenziale delle curve e delle superfici} - \begin{itemize} - \item $B_R(P, \RR^n)$, $B_R(P)$ -- la palla $n$-dimensionale di raggio $R$ e centro $P$. Ometteremo $\RR^n$ quando la dimensione si deduce dal contesto. - \item $S_a^i(P)$ -- l'ipersfera $i$-dimensionale di raggio $a$ e centro $P$, ovverosia $\{\vec{x} \in \RR^{i+1} \mid \norm{\vec{x} - P} = a\}$. - \item $\TT_{a, b}$ -- toro di raggio maggiore $a$ e raggio minore $b$. - \item $\ell(\alpha)$ -- lunghezza di una curva $\alpha$. - \item p.l.a.~-- parametrizzata a lunghezza d'arco, ovverosia con velocità unitaria. - \item $\kappa_\alpha$ -- curvatura di una curva $\alpha$ in un punto. - \item $\tau_\alpha$ -- torsione di una curva $\alpha$ in un punto. - \item $T_\alpha$ -- versore tangente di una curva regolare in un punto. - \item $N_\alpha$ -- versore normale di una curva di Frenet in un punto. - \item $B_\alpha$ -- versore binormale di una curva di Frenet in un punto. - \item $\Pi_\alpha$ -- piano osculatore di $\alpha$ in un punto. - \item $R_\alpha$ -- raggio di curvatura di $\alpha$ in un punto. - \item $T_P \Sigma$ -- piano tangente di $P$ rispetto a $\Sigma$. - \item $\vec{n}$, $n_{\vec{x}}$ -- (versore) normale (eventualmente locale) di una superficie o di una parametrizzazione regolare. - \item $D_\xi f(P)$ -- derivata direzionale con direzione $\xi$ della funzione $f$ nel punto $P$ di una superficie. - \item $S_P$ -- operatore forma nel punto $P$ di una superficie. - \item $\I_P$ -- I forma fondamentale nel punto $P$ di una superficie. - \item $\II_P$ -- II forma fondamentale nel punto $P$ di una superficie. - \item $E$, $F$, $G$ -- elementi della rappresentazione matriciale della I forma fondamentale: $\I_P = \begin{pmatrix} - E & F \\ F & G - \end{pmatrix}$. - \item $\ell$, $m$, $n$ -- elementi della rappresentazione matriciale della II forma fondamentale: $\II_P = \begin{pmatrix} - \ell & m \\ m & n - \end{pmatrix}$. - \item $a$, $b$, $c$, $d$ -- elementi della rappresentazione matriciale dell'operatore forma: $S_P = \begin{pmatrix} - a & c \\ b & d - \end{pmatrix}$. - \item $\kappa_n$ -- curvatura normale di una superficie in un punto $P$ secondo un dato vettore unitario. - \item $\kappa_{\alpha, n}$ -- curvatura normale di $\alpha$ rispetto a una superficie. - \item $\kappa_1$, $\kappa_2$ -- curvature principali di una superficie in un punto. - \item $\kappa$ -- curvatura gaussiana di una superficie in un punto. - \item $H$ -- curvatura media di una superficie in un punto. - \item $\Gamma_{ij}^k$ -- simbolo di Christoffel, ovverosia coefficiente di $\vec{x_k}$ in $\vec{x_{ij}}$. - \item $\nabla_v X(P)$ -- derivata covariante di un campo vettoriale $X$ tangente alla superficie $\Sigma$ in direzione $v$ nel punto $P$. - \item $(\cdot)^\top$ -- proiezione di $\cdot$ sul piano tangente $T_P \Sigma$. - \item $\gamma_v$ -- geodetica locale di direzione $v$. - \item $U_P$ -- intorno di definizione della mappa esponenziale in un punto $P$. - \item $\exp_P$ -- mappa esponenziale in un punto $P$. - \item $v_k$ -- data una base ortonormale $\vec{e_1}$, $\vec{e_2}$, la mappa $t \mapsto k(\cos(t) \vec{e_1} + \sin(t) \vec{e_2})$. - \item $N_P$ -- intorno normale di un punto $P$. - \item $\varphi(\gamma(t))$ -- angolo tra la curva $\gamma$ e il parallelo di $\gamma(t)$ al tempo $t$. - \item $r(\gamma(t))$ -- distanza di $\gamma(t)$ dall'asse di rotazione. - \item $k_{\alpha, g}$ -- curvatura geodetica di una curva $\alpha$ rispetto a una superficie $\Sigma$. - \item $A(R)$ -- area di una regione $R$ di una superficie $\Sigma$. - \item $\int_R \varphi \dA$ -- integrazione rispetto all'area in una regione $R$, equivalente a $\iint_{\vec{x}\inv(R)} (\varphi \circ \vec{x}) \, \norm{\vec{x_u} \times \vec{x_v}} \du \dv$. - \item $\eps_i$ -- angolo esterno $i$-esimo di una regione o superficie con bordo. - \item $\iota_i$ -- angolo interno $i$-esimo di una regione o superficie con bordo. - \item $\partial \Sigma$ -- bordo di una superficie con bordo, come unione delle tracce delle regioni con cui è definita per differenza. - \item $\chi(\Sigma)$ -- caratteristica di Eulero di una superficie con bordo. - \end{itemize} + \begin{itemize} + \item $B_R(P, \RR^n)$, $B_R(P)$ -- la palla $n$-dimensionale di raggio $R$ e centro $P$. Ometteremo $\RR^n$ quando la dimensione si deduce dal contesto. + \item $S_a^i(P)$ -- l'ipersfera $i$-dimensionale di raggio $a$ e centro $P$, ovverosia $\{\vec{x} \in \RR^{i+1} \mid \norm{\vec{x} - P} = a\}$. + \item $\TT_{a, b}$ -- toro di raggio maggiore $a$ e raggio minore $b$. + \item $\ell(\alpha)$ -- lunghezza di una curva $\alpha$. + \item p.l.a.~-- parametrizzata a lunghezza d'arco, ovverosia con velocità unitaria. + \item $\kappa_\alpha$ -- curvatura di una curva $\alpha$ in un punto. + \item $\tau_\alpha$ -- torsione di una curva $\alpha$ in un punto. + \item $T_\alpha$ -- versore tangente di una curva regolare in un punto. + \item $N_\alpha$ -- versore normale di una curva di Frenet in un punto. + \item $B_\alpha$ -- versore binormale di una curva di Frenet in un punto. + \item $\Pi_\alpha$ -- piano osculatore di $\alpha$ in un punto. + \item $R_\alpha$ -- raggio di curvatura di $\alpha$ in un punto. + \item $T_P \Sigma$ -- piano tangente di $P$ rispetto a $\Sigma$. + \item $\vec{n}$, $n_{\vec{x}}$ -- (versore) normale (eventualmente locale) di una superficie o di una parametrizzazione regolare. + \item $D_\xi f(P)$ -- derivata direzionale con direzione $\xi$ della funzione $f$ nel punto $P$ di una superficie. + \item $S_P$ -- operatore forma nel punto $P$ di una superficie. + \item $\I_P$ -- I forma fondamentale nel punto $P$ di una superficie. + \item $\II_P$ -- II forma fondamentale nel punto $P$ di una superficie. + \item $E$, $F$, $G$ -- elementi della rappresentazione matriciale della I forma fondamentale: $\I_P = \begin{pmatrix} + E & F \\ F & G + \end{pmatrix}$. + \item $\ell$, $m$, $n$ -- elementi della rappresentazione matriciale della II forma fondamentale: $\II_P = \begin{pmatrix} + \ell & m \\ m & n + \end{pmatrix}$. + \item $a$, $b$, $c$, $d$ -- elementi della rappresentazione matriciale dell'operatore forma: $S_P = \begin{pmatrix} + a & c \\ b & d + \end{pmatrix}$. + \item $\kappa_n$ -- curvatura normale di una superficie in un punto $P$ secondo un dato vettore unitario. + \item $\kappa_{\alpha, n}$ -- curvatura normale di $\alpha$ rispetto a una superficie. + \item $\kappa_1$, $\kappa_2$ -- curvature principali di una superficie in un punto. + \item $\kappa$ -- curvatura gaussiana di una superficie in un punto. + \item $H$ -- curvatura media di una superficie in un punto. + \item $\Gamma_{ij}^k$ -- simbolo di Christoffel, ovverosia coefficiente di $\vec{x_k}$ in $\vec{x_{ij}}$. + \item $\nabla_v X(P)$ -- derivata covariante di un campo vettoriale $X$ tangente alla superficie $\Sigma$ in direzione $v$ nel punto $P$. + \item $(\cdot)^\top$ -- proiezione di $\cdot$ sul piano tangente $T_P \Sigma$. + \item $\gamma_v$ -- geodetica locale di direzione $v$. + \item $U_P$ -- intorno di definizione della mappa esponenziale in un punto $P$. + \item $\exp_P$ -- mappa esponenziale in un punto $P$. + \item $v_k$ -- data una base ortonormale $\vec{e_1}$, $\vec{e_2}$, la mappa $t \mapsto k(\cos(t) \vec{e_1} + \sin(t) \vec{e_2})$. + \item $N_P$ -- intorno normale di un punto $P$. + \item $\varphi(\gamma(t))$ -- angolo tra la curva $\gamma$ e il parallelo di $\gamma(t)$ al tempo $t$. + \item $r(\gamma(t))$ -- distanza di $\gamma(t)$ dall'asse di rotazione. + \item $k_{\alpha, g}$ -- curvatura geodetica di una curva $\alpha$ rispetto a una superficie $\Sigma$. + \item $A(R)$ -- area di una regione $R$ di una superficie $\Sigma$. + \item $\int_R \varphi \dA$ -- integrazione rispetto all'area in una regione $R$, equivalente a $\iint_{\vec{x}\inv(R)} (\varphi \circ \vec{x}) \, \norm{\vec{x_u} \times \vec{x_v}} \du \dv$. + \item $\eps_i$ -- angolo esterno $i$-esimo di una regione o superficie con bordo. + \item $\iota_i$ -- angolo interno $i$-esimo di una regione o superficie con bordo. + \item $\partial \Sigma$ -- bordo di una superficie con bordo, come unione delle tracce delle regioni con cui è definita per differenza. + \item $\chi(\Sigma)$ -- caratteristica di Eulero di una superficie con bordo. + \end{itemize} - \section*{Teoria della misura} - \addcontentsline{toc}{section}{Teoria della misura} + \section*{Teoria della misura} + \addcontentsline{toc}{section}{Teoria della misura} - \begin{itemize} - \item $\vol$ -- applicazione per calcolare il volume di un rettangolo in $\RR^n$; tale per cui - $\vol([a_1, b_1] \times \cdots \times [a_n, b_n]) = \prod_i {(b_i - a_i)}$. - \item $m$ -- misura di Lebesgue su $\RR^n$. - \end{itemize} + \begin{itemize} + \item $\vol$ -- applicazione per calcolare il volume di un rettangolo in $\RR^n$; tale per cui + $\vol([a_1, b_1] \times \cdots \times [a_n, b_n]) = \prod_i {(b_i - a_i)}$. + \item $m$ -- misura di Lebesgue su $\RR^n$. + \end{itemize} - \section*{Teoria delle varietà} - \addcontentsline{toc}{section}{Teoria delle varietà} + \section*{Teoria delle varietà} + \addcontentsline{toc}{section}{Teoria delle varietà} - \begin{itemize} - \item $\dim M$ -- dimensione di una varietà $M$. - \item $(f, W \cap M)$ -- carta locale di una varietà $M$; si sottintende che $W$ sia un aperto dello spazio ambiente in cui è contenuto $M$ e - che $f$ sia un diffeomorfismo con dominio $W \cap M$ verso un aperto di $\RR^{\dim M}$. - \item $T_x M$ -- spazio tangente di un punto $x$ di una varietà $M$. - \item $\dif f_x$ -- differenziale di una mappa liscia $f : M \to N$ tra varietà nel punto $x$. - \item $\crit(f)$ -- insieme dei punti critici di una mappa liscia $f : M \to N$ tra varietà. - \item $S^n$ -- sfera $n$-dimensionale reale, ovverosia $\{\vec{x} \in \RR^{n+1} \mid \norm{\vec{x}} = 1\}$. - \item $D^n$ -- disco $n$-dimensionale reale, ovverosia $\{\vec{x} \in \RR^{n} \mid \norm{\vec{x}} \leq 1\}$. - \item $H^n$ -- semispazio $\{ x \in \RR^n \mid x_m \geq 0 \}$. - \item $\partial H^n$ -- bordo del semispazio $\{ x \in \RR^n \mid x_m \geq 0 \}$, ovverosia $\{ x \in \RR^n \mid x_m = 0 \} \cong \RR^{n-1}$. - \item $\partial M$ -- bordo di una varietà bordata $M$. - \end{itemize} + \begin{itemize} + \item $\dim M$ -- dimensione di una varietà $M$. + \item $(f, W \cap M)$ -- carta locale di una varietà $M$; si sottintende che $W$ sia un aperto dello spazio ambiente in cui è contenuto $M$ e + che $f$ sia un diffeomorfismo con dominio $W \cap M$ verso un aperto di $\RR^{\dim M}$. + \item $T_x M$ -- spazio tangente di un punto $x$ di una varietà $M$. + \item $\dif f_x$ -- differenziale di una mappa liscia $f : M \to N$ tra varietà nel punto $x$. + \item $\crit(f)$ -- insieme dei punti critici di una mappa liscia $f : M \to N$ tra varietà. + \item $S^n$ -- sfera $n$-dimensionale reale, ovverosia $\{\vec{x} \in \RR^{n+1} \mid \norm{\vec{x}} = 1\}$. + \item $D^n$ -- disco $n$-dimensionale reale, ovverosia $\{\vec{x} \in \RR^{n} \mid \norm{\vec{x}} \leq 1\}$. + \item $H^n$ -- semispazio $\{ x \in \RR^n \mid x_m \geq 0 \}$. + \item $\partial H^n$ -- bordo del semispazio $\{ x \in \RR^n \mid x_m \geq 0 \}$, ovverosia $\{ x \in \RR^n \mid x_m = 0 \} \cong \RR^{n-1}$. + \item $\partial M$ -- bordo di una varietà bordata $M$. + \end{itemize} - \section*{Topologia} - \addcontentsline{toc}{section}{Topologia} + \section*{Topologia} + \addcontentsline{toc}{section}{Topologia} - \begin{itemize} - \item II-numerabile -- spazio topologico che ammette una base numerabile. - \item T1 -- spazio topologico i cui singoletti sono insiemi chiusi. - \item T2 -- spazio topologico per cui due punti distinti ammettono l'esistenza di una coppia di intorni disgiunti. - \end{itemize} + \begin{itemize} + \item II-numerabile -- spazio topologico che ammette una base numerabile. + \item T1 -- spazio topologico i cui singoletti sono insiemi chiusi. + \item T2 -- spazio topologico per cui due punti distinti ammettono l'esistenza di una coppia di intorni disgiunti. + \item $\overline{X}$ -- chiusura di un insieme. + \item $\Int(X)$ -- parte interna di un insieme. + \item $\partial X$ -- frontiera di un insieme, ovverosia $\overline{X} \setminus \Int(X)$. + \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 9ec4cb1..621ad62 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 @@ -4,9 +4,17 @@ \setlength{\parindent}{2pt} \begin{multicols*}{2} + \section*{Algebra lineare} + \addcontentsline{toc}{section}{Algebra lineare} - \section*{Analisi matematica e teoria della misura} - \addcontentsline{toc}{section}{Analisi matematica e teoria della misura} + \begin{itemize} + \item \textbf{Caratterizzazione del rango di una matrice} -- Sia $M \in \RR^{m \times n}$ una matrice. Allora $\rk(M) = k$ se e solo se + i minori di taglia superiore a $k$ in $M$ hanno tutti determinante nullo ed esiste un minore di + taglia $k$ con determinante \underline{non} nullo. + \end{itemize} + + \section*{Analisi matematica} + \addcontentsline{toc}{section}{Analisi matematica} \begin{itemize} \item \textbf{Teorema di Schwarz} -- Sia $f : \RR^n \to \RR$ una funzione che ammette derivate seconde @@ -72,6 +80,17 @@ \] è un aperto di $\RR \times \Omega$ e l'applicazione $\Phi : \mathcal{D} \to \RR^n$ è di classe $C^k$. + \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. + + \item \textbf{Teorema di approssimazione di Weierstrass} -- Sia $K \subseteq \RR^n$ un compatto, e sia $f : K \to \RR$ una mappa + continua. Allora per ogni $\eps > 0$ esiste una funzione polinomiale $P : K \to \RR$ tale per cui $\norm{f-P}_\infty < \eps$. + \end{itemize} + + \section*{Teoria della misura} + \addcontentsline{toc}{section}{Teoria della misura} + + \begin{itemize} \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: @@ -80,11 +99,6 @@ \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. - - \item \textbf{Teorema di approssimazione di Weierstrass} -- Sia $K \subseteq \RR^n$ un compatto, e sia $f : K \to \RR$ una mappa - continua. Allora per ogni $\eps > 0$ esiste una funzione polinomiale $P : K \to \RR$ tale per cui $\norm{f-P}_\infty < \eps$. \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 4775423..c6f7da0 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 @@ -337,6 +337,27 @@ $f$ è esattamente $f(\crit(f))$. \end{remark} + \begin{remark}[I punti regolari formano un aperto] \label{rmk:punti_regolari_formano_aperto} + Se $x$ è un punto regolare di una mappa liscia + $f : M \to N$, esiste sempre un intorno aperto $U$ + di $x$ in $M$ composto di soli punti regolari. \smallskip + + Scelta una parametrizzazione locale $g : U \to g(U)$ di un intorno aperto di $x$, + si può scegliere infatti una base ``comune'' per ogni $T_y M$ al + variare di $y$ in $g(U)$, e così si può rappresentare + $\dif f_y$ matricialmente. \smallskip + + Poiché $x$ è regolare, $\dif f_x$ è surgettiva. Allora + $\dif f_x$ ammette un minore di taglia massima di + determinante \underline{non} nullo. Il determinante di questo minore, + al variare di $y \in g(U)$, varia continuamente; in particolare, per + il Teorema della permanenza del segno, esiste un intorno di $x$ + in cui continua a essere \underline{non} nullo. \smallskip + + Equivalentemente esiste un intorno aperto di $x$ in cui tutti i punti + sono regolari. + \end{remark} + \subsection{Teorema di invertibilità locale per varietà e lemma della pila di dischi} \begin{theorem}[di invertibilità locale per varietà] \label{thm:invertibilità_locale_varietà} @@ -537,7 +558,7 @@ per cui $f \circ \iota$ è costante. Tuttavia una mappa costante ha differenziale nullo, e dunque: \[ - \dif f_x \circ \underbrace{\iota_{T_x P}}_{= \dif \iota^P_x} = 0 \implies T_x P \subseteq \ker(\dif f_x). + \dif f_x \circ \underbrace{\iota_{T_x P}}_{\mathclap{= \, \dif \iota^P_x}} = 0 \implies T_x P \subseteq \ker(\dif f_x). \] Poiché $\dim T_x P = m - n = \ker(\dif f_x)$, si ottiene l'uguaglianza $T_x P = \ker(\dif f_x)$. \smallskip @@ -751,7 +772,7 @@ \subsection{Varietà con bordo da valori regolari} - \begin{theorem} \label{lem:varietà_bordata_da_valore_regolare_su_R} + \begin{theorem} \label{thm:varietà_bordata_da_valore_regolare_su_R} Sia $f : M \to \RR$ una mappa liscia tra varietà, dove $M$ è una $m$-varietà senza bordo. Se $0$ è un valore regolare per $f$, allora $\{ f \geq 0 \}$ è una $m$-varietà con bordo $f\inv(0)$. @@ -781,7 +802,7 @@ \end{proof} \begin{remark} - Chiaramente il Lemma \ref{lem:varietà_bordata_da_valore_regolare_su_R} si generalizza + Chiaramente il Teorema \ref{thm:varietà_bordata_da_valore_regolare_su_R} si generalizza a qualsiasi insieme della forma $\{ f \operatorname{op} a \}$ con $\operatorname{op} \in \{\leq, \geq\}$ e $a$ valore regolare di $f$. \end{remark} @@ -797,21 +818,74 @@ \] Allora, come visto per il Corollario \ref{cor:sn_è_varietà}, $1$ è valore regolare di $f$, e quindi $D^n = \{ f \leq 1 \}$ è - una $n$-varietà con bordo $S^{n-1}$. + una $n$-varietà con bordo $S^{n-1}$ per il Teorema \ref{thm:varietà_bordata_da_valore_regolare_su_R}. \end{proof} \begin{lemma} \label{lem:varietà_bordata_da_valore_regolare_caso_particolare} - Sia $f : H^m \to \RR^n$ con $m > n$ una mappa liscia. Se $y \in \RR^n$ è un valore regolare + Sia $f : H^m \to \RR^n$ con $m > n$ una mappa liscia. \smallskip + + Se $y \in \RR^n$ è un valore regolare sia per $f$ che per $\restr{f}{\partial H^m}$ con $f\inv(y) \neq \emptyset$, allora $f\inv(y)$ è una $(m-n)$-varietà con bordo $\partial f\inv(y) = f\inv(y) \cap \partial H^m$. \end{lemma} \begin{proof} - ... + Sia $x \in f\inv(y)$. Supponiamo valga $x \in \Int(H^m) = H^m \setminus \partial H^m$. Possiamo + restringerci a un intorno aperto $U$ di $x$ in $\RR^m$, per il quale $\restr{f}{U}$ diventa una mappa tra + varietà senza bordo. Allora $\restr{f}{U}\inv(y)$ è una $(m-n)$ varietà senza bordo per il + Teorema \ref{thm:varietà_da_valore_regolare}. Quindi $x$ eredita da questa varietà le carte locali su $f\inv(y)$; + inoltre $x$ \underline{non} potrà appartenere al bordo di $f\inv(y)$, essendo nell'immagine di una parametrizzazione + di un aperto di $\RR^{m-n}$. \smallskip + + Supponiamo valga adesso $x \in \partial H^m$. Poiché $f$ è liscia, esistono un intorno aperto $W$ + di $x$ in $\RR^m$ e un'estensione liscia $F : W \to \RR^n$ per cui: + \[ + \restr{F}{W \cap H^m} = \restr{f}{W \cap H^m}. + \] + Dal momento che $\dif F_x = \dif f_x$, e $x$ è un punto regolare per $f$, allora $\dif F_x$ è surgettiva, + e $x$ è punto regolare anche per $F$. Dal momento che i punti regolari formano un aperto (vd. Osservazione + \ref{rmk:punti_regolari_formano_aperto}), possiamo supporre, a meno di restringere $W$, che \underline{non} + vi siano punti critici in $W$. Pertanto, $y$ sarà valore regolare per $F$ e $F\inv(y)$ è dunque una $(m-n)$-varietà + senza bordo per il Teorema \ref{thm:varietà_da_valore_regolare}. \smallskip + + Sia $\pi : \RR^m \to \RR$ la proiezione tale per cui $x \mapsto x_m$. Allora $\pi$ è una mappa liscia. \smallskip + + Mostriamo che $0$ è un valore regolare per $\restr{\pi}{F\inv(y)}$. Osserviamo innanzitutto che: + \[ + (\restr{\pi}{F\inv(y)})\inv(0) = F\inv(y) \cap \partial H^m \underbrace{=}_{\mathclap{(W \setminus H^m) \cap \partial H^m = \emptyset}} f\inv(y) \cap \partial H^m. + \] + Sia $y^* \in f\inv(y) \cap \partial H^m$. Osserviamo che: + \begin{enumerate}[(i.)] + \item $T_{y^*} F\inv(y) = \ker \dif F_{y^*} = \ker \dif f_{y^*}$ per la Proposizione \ref{prop:tangente_valore_regolare}, e questo spazio + ha dimensione $m-n$. + \item $T_{y^*} \partial H^m = \ker \dif \pi_{y^*}$ per la Proposizione \ref{prop:tangente_valore_regolare}. + \item $\ker \dif (\restr{f}{\partial H^m})_{y^*} = \ker \restr{\dif f_{y^*}}{T_{y^*} \partial H^m}$ ha dimensione + $m-n-1$ dal momento che $y$ è per ipotesi un valore regolare di $\restr{f}{\partial H^m}$, dove per l'uguaglianza + dei due nuclei si è utilizzato che: + \[ + \restr{f}{\partial H^m} = f \circ \iota^{\partial H^m}, + \] + e successivamente la \textit{chain rule}. + \item Grazie a (i.), (ii.) e (iii.), possiamo scrivere: + \[ + \ker \dif (\restr{f}{\partial H^m})_{y^*} = T_{y^*} F\inv(y) \cap (\ker \dif \pi_{y^*}). + \] + \end{enumerate} + + Osserviamo che $y^*$ è un punto critico per $\restr{\pi}{F\inv(y)}$ se e solo se $\dif (\restr{\pi}{F\inv(y)})_{y^*}$ è nullo, + dacché $\pi$ è una mappa in una $1$-varietà. + Questo accade se e solo se $T_{y^*} F\inv(y) \subseteq \ker \dif \pi_{y^*}$. Tuttavia, se vi fosse questa inclusione, si avrebbe per (iv.) + $\ker \dif (\restr{f}{\partial H^m})_{y^*} = T_{y^*} F\inv(y)$, che è assurdo dal momento che il primo spazio ha dimensione $m-n-1$ per (iii.) + e il secondo ha dimensione $m-n$ per (i.). Quindi $0$ è regolare per $\restr{\pi}{F\inv(0)}$. \smallskip + + Dal momento che $0$ è regolare per $\restr{\pi}{F\inv(0)}$, per il Teorema \ref{thm:varietà_bordata_da_valore_regolare_su_R} + $\{\pi \geq 0\} = f\inv(y) \cap H^m$ è una $(m-n)$-varietà con bordo $\{\pi = 0\} = f\inv(y) \cap \partial H^m \ni x$. + Quindi $x$ eredita da questa varietà le carte locali su $f\inv(y)$; inoltre $x$ appartiene al bordo di $f\inv(y)$, + essendo immagine di un punto di bordo tramite una parametrizzazione locale. \end{proof} \begin{theorem} \label{thm:varietà_bordata_da_valore_regolare} - Sia $f : M \to N$ una mappa liscia tra varietà, dove $M$ è una $m$-varietà, + Sia $f : M \to N$ una mappa liscia tra varietà, dove $M$ è una $m$-varietà con bordo $\partial M$ \underline{non} vuoto, $N$ è una $n$-varietà senza bordo e $m > n$. \smallskip Se $y \in N$ è un valore regolare sia per $f$ che per $\restr{f}{\partial M}$ con $f\inv(y) \neq \emptyset$,