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zY&nUvr?601)W6! zaPsDxwYM4Fpn?H@0jkyW)iOudU-Azw4klr?mvMhjx@F>A!@9*e*t%ef zlKezT%%UriP}^)GflvlNVg{J+H|)~oAH_Eu~7 zeZsw#c4H0r*2m!M87bA|6UK3eY7vL4;g&z9{#jSi8tmQmc^;X4#Zrj*CmJz*G3z>i zM1n3=CIU(~VFdqO)U~{7 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 $\mult(p, z)$ -- molteplicità algebrica della radice $z$ nel polinomio $p$. + \item $\deg(p(x))$ -- grado del polinomio $p$. + \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-Poincaré 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 $M$. + \item $\deg_2 f$ -- grado modulo $2$ di una mappa liscia $f : M \to N$, con $M$ compatta e $N$ connessa. + \item $\sgn$ -- segno di un'orientazione rispetto a una ``standard''; per una mappa lineare, funzione che restituisce + $+1$ in caso di orientazione preservata e $-1$ altrimenti. + \item $\deg(f; y)$ -- grado intero di $f$ in $y$, dove $f : M \to N$ è una mappa liscia con $M$ chiusa e orientata, e $N$ connessa e orientata. + \item $\deg f$ -- grado intero di una mappa liscia $f : M \to N$, con $M$ chiusa e orientata, e $N$ connessa e orientata. + \item $\ind(v, z)$ -- indice del campo vettoriale $v$ in $z$. + \item $\Delta^{(m)}$ -- $m$-simplesso. + \item $s_i(C)$ -- numero di $i$-simplessi nel complesso simpliciale $C$. + \item $\chi(M)$ -- caratteristica di Eulero-Poincaré di una varietà compatta $M$. + \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. - \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} + \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/sections/3-curve_su_superfici.tex b/Corsi/Geometria e topologia differenziale/Scheda/sections/3-curve_su_superfici.tex index c480350..5750748 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda/sections/3-curve_su_superfici.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda/sections/3-curve_su_superfici.tex @@ -1171,16 +1171,16 @@ \end{itemize} \end{definition} - \subsection{Teorema di Radó e caratteristica di Eulero} + \subsection{Teorema di Radó e caratteristica di Eulero-Poincaré} \begin{theorem}[Radó] \label{thm:rado} Ogni superficie orientata con bordo ammette una triangolarizzazione. \end{theorem} - \begin{definition}[Caratteristica di Eulero] + \begin{definition}[Caratteristica di Eulero-Poincaré] Sia $\tau = \{ \Delta_\lambda \}_{\lambda=1}^n$ una triangolarizzazione di una superficie orientata con bordo. Allora si definisce la sua - \textbf{caratteristica di Eulero} come: + \textbf{caratteristica di Eulero-Poincaré} come: \[ \boxed{\chi(\Sigma) \defeq V - L + T,} \] @@ -1189,7 +1189,7 @@ \end{definition} \begin{fact} - La caratteristica di Eulero \underline{non} dipende dalla + La caratteristica di Eulero-Poincaré \underline{non} dipende dalla triangolarizzazione scelta. \end{fact}