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b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex index 11788b0..9f473e9 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex @@ -32,7 +32,7 @@ \renewcommand{\vec}[1]{{\underaccent{\bar}{{#1}}}} -\newtheorem*{warn}{\warning \; Attenzione} +\newtheorem*{warn}{{\fontencoding{U}\fontfamily{futs}\selectfont\char 49\relax} \; Attenzione} \newtheoremstyle{customth} {\topsep}{\topsep} @@ -99,6 +99,8 @@ \newcommand{\der}[2]{\frac{\mathop{}\!\textnormal{\slshape d} #1}{\mathop{}\!\textnormal{\slshape d} #2}} \newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\dertime}[1]{\frac{\mathop{}\!\textnormal{\slshape d}}{\mathop{}\!\textnormal{\slshape d} t} \bigg|_{#1}} + \newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} \newcommand{\dx}{\dif{x}} 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 463b086..9119f79 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 @@ -9,6 +9,10 @@ \addcontentsline{toc}{section}{Analisi matematica} \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 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 6806829..9dbc816 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 @@ -161,7 +161,7 @@ a ogni tempo $s$ il \textbf{versore normale} $N_\beta(s)$ così definito: \[ - N_\beta(s) = \frac{T_\beta(s)}{\norm{T_\beta(s)}}. + N_\beta(s) \defeq \frac{T_\beta(s)}{\norm{T_\beta(s)}}. \] \end{definition} @@ -170,7 +170,7 @@ a ogni tempo $s$ il \textbf{versore binormale} $B_\beta(s)$ così definito: \[ - B_\beta(s) = T_\beta(s) \times N_\beta(s). + B_\beta(s) \defeq T_\beta(s) \times N_\beta(s). \] \end{definition} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/2-superfici.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/2-superfici.tex index 7a4dfae..68fa75a 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/2-superfici.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/2-superfici.tex @@ -91,12 +91,20 @@ Sia $\alpha : I \to \RR^3$ una curva parametrizzata della forma $(a(t), 0, b(t))$ tale che $\alpha$ è regolare e omeomorfismo locale. Si definisce allora la \textbf{superficie di rotazione} (intorno - all'asse $z$) di $\alpha$ come l'immagine della seguente parametrizzazione: + all'asse $z$) di $\alpha$ come l'immagine della seguente + parametrizzazione canonica: \[ \vec{x}(u, v) = (a(u) \cos(v), a(u) \sin(v), b(u)). \] \end{definition} + \begin{definition}[Paralleli e meridiani] + Sia $\Sigma$ una superficie di rotazione con parametrizzazione canonica + $\vec{x}$. Allora l'immagine della curva $\alpha_{u_0}(t) = \vec{x}(u_0, t)$ + è detta \textbf{parallelo}, mentre quella della curva + $\gamma_{v_0}(t) = \vec{x}(t, v_0)$ è detta \textbf{meridiano}. + \end{definition} + \begin{proposition} Una superficie di rotazione è effettivamente una superficie, poiché la sua parametrizzazione canonica è regolare. @@ -242,4 +250,10 @@ Il gradiente $\nicefrac{\nabla f}{\norm{\nabla f}}$ è un campo vettoriale unitario e ortogonale a $T_P \Sigma$ per ogni punto $P$. Si conclude per la Proposizione \ref{prop:normale_continua}. \end{proof} + + \begin{warn} + Ogni superficie è localmente orientabile! \\[0.5em] + È sufficiente prendere per ogni + punto come ricoprimento la sua stessa parametrizzazione regolare. + \end{warn} \end{multicols*} \ No newline at end of file diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex new file mode 100644 index 0000000..65600f5 --- /dev/null +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex @@ -0,0 +1,90 @@ +%-------------------------------------------------------------------- +\chapter{Curve su superfici} +\setlength{\parindent}{2pt} + +\begin{multicols*}{2} + \section{Piano tangente e derivata direzionale} + + \subsection{Coordinate di una curva rispetto a una parametrizzazione regolare} + + \begin{proposition} \label{prop:coordinate_curva_parametrizzazione} + Sia $\vec{x} : U \to \RR^3$ una parametrizzazione regolare e sia $\alpha : I \to \vec{x}(U)$ + una curva. Allora $\alpha$ si scrive come: + \[ + \alpha(t) = \vec{x}(u(t), v(t)), + \] + con $u(t)$, $v(t) : I \to U$ funzioni di classe $C^\infty$. + \end{proposition} + + \begin{proof} + Segue immediatamente dalla Proposizione \ref{prop:parametrizzazione_è_cinf}. + \end{proof} + + \subsection{Relazione tra il piano tangente e le velocità delle curve} + + \begin{proposition}[Il piano tangente è l'insieme delle velocità delle curve sulla superficie considerata] + Sia $\Sigma$ una superficie. Allora vale: + \[ + T_P \Sigma = \{ \alpha'(P) \mid \alpha : I \to \Sigma \textnormal{ curva con } P \in \alpha(I) \}. + \] + \end{proposition} + + \begin{proof} + Sia $P = \vec{x}(0, 0)$, dove $\vec{x} : B_\eps \to \Sigma$ è una parametrizzazione regolare di $P$. + È sufficiente osservare che ogni vettore tangente è una combinazione lineare della forma + $\lambda \vec{x_u} + \mu \vec{x_v}$; allora la curva + $\alpha(t) = \vec{x}(t \lambda, t \mu)$ ha velocità $\lambda \vec{x_u} + \mu \vec{x_v}$ in + $P$. + \end{proof} + + \subsection{Funzioni lisce sulla superficie e derivata direzionale} + + \begin{definition}[Funzioni $C^\infty$ sulla superficie] + Sia $\Sigma$ una superficie. Una funzione $f : \Sigma \to \RR^n$ si + dice di classe $C^\infty$ se per ogni parametrizzazione regolare + $\vec{x}$ di ogni punto $P \in \Sigma$, $f \circ \vec{x}$ è di + classe $C^\infty$. + \end{definition} + + \begin{proposition} + Sia $\Sigma$ una superficie. Sia $P \in \Sigma$ e venga dato + $\xi \in T_P \Sigma$. Sia data una funzione $f : \Sigma \to \RR^n$. + Se $\alpha$, $\beta$ sono due curve su $\Sigma$ + passanti per $P$ al tempo $0$ con $\alpha'(0) = \beta'(0) = \xi$, allora: + \[ + (f \circ \alpha)'(0) = (f \circ \beta)'(0). + \] + \end{proposition} + + \begin{proof} + Sia $\vec{x}$ una parametrizzazione regolare di $P$ rispetto a $\Sigma$. + Allora, per la Proposizione \ref{prop:coordinate_curva_parametrizzazione}, + $\alpha(t) = (u(t), v(t))$ e $\beta(t) = (p(t), q(t))$, con $u$, $v$, $p$, $q$ lisce. \smallskip + + Pertanto: + \[ + \begin{aligned} + (f \circ \alpha)'(0) & = \dertime{t=0} (f \circ x)(u(t), v(t)) \\[1em] + & = (f \circ x)_u u'(0) + (f \circ x)_v v'(0). + \end{aligned} + \] + + Osserviamo che $(u'(0), v'(0)) = (p'(0), q'(0))$, dal momento che rappresentano le coordinate + in $U$ del vettore $\xi$. + La tesi segue allora dal fatto + che il membro a destra diventa $(f \circ \beta)'(0)$. + \end{proof} + + \begin{definition}[Derivata direzionale] + Sia $\Sigma$ una superficie. Sia $P \in \Sigma$ e venga dato + $\xi \in T_P \Sigma$. Allora, data una funzione $f : \Sigma \to \RR^n$, + la derivata direzionale $D_\xi f(P)$ è definita come: + \[ + D_\xi f(P) \defeq (f \circ \alpha)'(0), + \] + dove $\alpha$ è una qualsiasi curva su $\Sigma$ + passante per $P$ al tempo $0$ con $\alpha'(0) = \xi$. + \end{definition} + + %\section{Operatore forma, I e II forma fondamentale} +\end{multicols*}