From 22525e4cd9da29dd148701eecb6d543651f62e6c Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Wed, 19 Apr 2023 10:05:52 +0200 Subject: [PATCH] feat(geometria): aggiunge la base degli appunti di geometria --- .../main.pdf | Bin 217649 -> 227476 bytes .../main.tex | 98 ++++++++++++- .../main.pdf | Bin 202583 -> 202575 bytes .../main.tex | 2 +- .../main.pdf | Bin 0 -> 206535 bytes .../main.tex | 138 ++++++++++++++++++ tex/latex/style/personal_commands.sty | 1 + 7 files changed, 234 insertions(+), 5 deletions(-) rename Analisi matematica 1/Parte teorica/{2023-03-31, 04, Teoria sulle derivate => 2023-03-31, 04, 18, Teoria sulle derivate}/main.pdf (58%) rename Analisi matematica 1/Parte teorica/{2023-03-31, 04, Teoria sulle derivate => 2023-03-31, 04, 18, Teoria sulle derivate}/main.tex (81%) create mode 100644 Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani/main.pdf create mode 100644 Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti 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sulle derivate/main.tex rename to Analisi matematica 1/Parte teorica/2023-03-31, 04, 18, Teoria sulle derivate/main.tex index 9187fd2..9a55b3c 100644 --- a/Analisi matematica 1/Parte teorica/2023-03-31, 04, Teoria sulle derivate/main.tex +++ b/Analisi matematica 1/Parte teorica/2023-03-31, 04, 18, Teoria sulle derivate/main.tex @@ -4,7 +4,7 @@ \title{\textbf{Note del corso di Analisi Matematica 1}} \author{Gabriel Antonio Videtta} -\date{31 marzo e 4 aprile 2023} +\date{31 marzo, 4 e 18 aprile 2023} \begin{document} @@ -287,6 +287,42 @@ f(d) \geq f(c)$, e quindi $f$ è crescente in $I$, da cui la tesi. \end{proof} + \begin{proposition} + Sia $I \subset \RR$ un intervallo e sia $f : I \to \RR$ tale che $f$ + sia derivabile in $I$. Allora $f$ è convessa + se e solo se la derivata è crescente. + \end{proposition} + + \begin{proof} Si dimostrano le due implicazioni separatamente. \\ + + \rightproof Siano $x_0$, $x_1 \in I$ con $x_0 < x_1$. Sia $h$ + positivo tale che $x_0 < x_0 + h \leq x_1$. Allora $x_0 + h = (1- \lambda) x_0 + \lambda x_1$ con $\lambda = \frac{h}{x_1 - x_0}$. + Allora, poiché $f$ è convessa, $f(x_0 + h) \leq (1- \lambda) f(x_0) + f(x_1) \leq f(x_0) + \frac{h}{x_1 - x_0} \left( f(x_1) - f(x_0) \right)$, da cui si ricava che: + + \[ \frac{f(x_0 + h) - f(x_0)}{h} \leq \frac{f(x_1) - f(x_0)}{x_1 - x_0}. \] + + Quindi, passando al limite, $f'(x_0) \leq \frac{f(x_1) - f(x_0)}{x_1 - x_0}$. Analogamente si dimostra che $\frac{f(x_1) - f(x_0)}{x_1 - x_0} \leq f'(x_1)$. Si conclude dunque che $f'(x_1) \geq f'(x_0)$, ossia + che $f'$ è crescente. \\ + + \leftproof Siano $x_0$, $x_1 \in I$ con $x_0 < x_1$. Si considera + $x = (1 - \lambda) x_0 + \lambda x_1 \in (x_0, x_1)$ con $0 < \lambda < 1$. Per il teorema di Lagrange $\exists \tilde{x_0} \in (x_0, x)$ tale + che $f'(\tilde{x_0}) = \frac{f(x) - f(x_0)}{x - x_0}$. Analogamente + $\exists \tilde{x_1} \in (x, x_1)$ tale che $f'(\tilde{x_1}) = + \frac{f(x_1) - f(x)}{x_1 - x}$. Poiché allora per ipotesi la + derivata $f'$ è crescente, si ricava che: + + \[ \frac{f(x_1) - f(x)}{x_1 - x} \geq \frac{f(x) - f(x_0)}{x - x_0}, \] + + da cui si conclude che: + + \[ f(x) \leq (1-\lambda) f(x_0) + \lambda f(x_1), \] + + \vskip 0.05in + + ossia che vale la disuguaglianza di Jensen, e quindi che + $f$ è convessa, da cui la tesi. + \end{proof} + \begin{remark}\nl \li L'interpretazione geometrica del teorema di Cauchy, rispetto a quella di Lagrange, è leggermente più complicata. Si consideri @@ -297,7 +333,11 @@ come $\frac{f'(\xbar)}{g'(\xbar)}$. Allora, il teorema di Cauchy asserisce che esiste un punto della curva $\gamma$ tale per cui la retta tangente alla curva in quel punto è parallela alla secante - passante per $(g(a), f(a))$ e $(g(b), f(b))$. + passante per $(g(a), f(a))$ e $(g(b), f(b))$. \\ + + \li Inoltre $f$ è strettamente crescente in $I$ se $f' \geq 0$ e + non esistono intervalli di punti stazionari. Analogamente se $f' < 0$ in + $I$ e non esistono ancora tali intervalli, $f$ è strettamente decrescente in $I$. \end{remark} \begin{exercise} @@ -336,7 +376,7 @@ in un intorno $J$ di $\xbar$, allora $f_1(x) + f_2(x) \tendsto{\xbar} +\infty$; \item se $f_1 \tendsto{\xbar} 0$ e $f_2$ è limitata in un intorno $J$ di $\xbar$, allora $f_1 f_2(x) \tendsto{\xbar} 0$; - \item se $f_1 \tendsto{\xbar} +\infty$ è limitata inferiormente + \item se $f_1 \tendsto{\xbar} +\infty$ ed $f_2$ è limitata inferiormente da una costante positiva $c$ in un intorno $J$ di $\xbar$, allora $f_1 f_2 \tendsto{\xbar} +\infty$. \end{enumerate} @@ -391,6 +431,56 @@ \[ D_+ f(0) = \lim_{h \to 0^+} \frac{h + 2h^2 \sin\left(\frac{1}{h}\right)}{h} = 1 + \lim_{h \to 0^+} 2 h \sin\left(\frac{1}{h}\right) = 1, \] - dove si è usato lo stesso argomento di prima per computare $\lim_{h \to 0^+} 2 h \sin\left(\frac{1}{h}\right) = 0$. \\ + dove si è usato lo stesso argomento di prima per computare $\lim_{h \to 0^+} 2 h \sin\left(\frac{1}{h}\right) = 0$. \end{solution} + + \begin{theorem} (di de l'Hopital) %TODO: sistemare nome + Siano $I$ intervallo e $x_0 \in I$. Sia detto $I' := I \setminus \{ x_0 \}$. Siano $f$, $g : I' \to \RR$ derivabili tali che: + + \begin{enumerate}[(i)] + \item esiste $L := \lim_{x \to x_0} \frac{f'(x)}{g'(x)}$, + \item $g' \neq 0$ in $I'$, + \item vale che (a) $f(x)$, $g(x) \tendsto{x_0} 0$ oppure che (b) + $g(x) \tendsto{x_0} \pm \infty$. + \end{enumerate} + + Allora $\frac{f(x)}{g(x)} \tendsto{x_0} L$. + \end{theorem} + + \begin{proof} + Si consideri il caso (a) per $x_0$ finito. Si ponga $f(x_0) = g(x_0) := 0$. Senza perdità di generalità si + assuma che $I$ sia un intorno destro di $x_0$. Sia $x \in I \setminus \{x_0\}$, da cui si ricava che $x > x_0$. \\ + + Si osserva che $\frac{f(x)}{g(x)} = \frac{f(x) - f(x_0)}{g(x)} - g(x_0)$. Per il teorema di Cauchy, esiste allora $\tilde{x} \in (x_0, x)$, + in funzione di $x$, tale + che $\frac{f(x)}{g(x)} = \frac{f'(\tilde x)}{g'(\tilde x)}$. Allora + $\lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f'(\tilde x)}{g'(\tilde x)} = \lim_{\tilde x \to x_0} \frac{f'(\tilde x)}{g'(\tilde x)} = L$, dove si è utilizzato che $\tilde x \tendsto{x_0} x_0$ per il teorema del confronto applicato sulla + relazione $x_0 < \tilde x < x$. \\ + + Si consideri ora il caso (b) per $x_0$ finito. Siano $x_1 > x_0$ + tali che $x_1 > x > x_0$. Allora vale la seguente identità: + + \[ \frac{f(x)}{g(x)} = \left( \frac{f(x) - f(x_1)}{g(x) - g(x_1)} + \frac{f(x_1)}{g(x) - g(x_1)} \right) \frac{g(x) - g(x_1)}{g(x)}. \] + + Si osserva allora che: + + \[ \lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f(x) - f(x_1)}{g(x) - g(x_1)}. \] + \end{proof} + + \begin{remark} + È essenziale che $I$ sia un intervallo affinché il teorema di de + l'Hopital sia vero. + % TODO: scrivi esempio di unione infinita di intervalli (tra x e x^2 + x) + \end{remark} + + \begin{proposition} + Sia $I$ un intervallo, sia $x_0 \in I$ e sia $f : I \to \RR$ continua + e derivabile dappertutto tranne che in $x_0$. Se esiste + $L := \lim_{x \to x_0} f'(x)$, allora $f'(x_0) = L$. + \end{proposition} + + \begin{proof} + Si consideri il rapporto incrementale $\frac{f(x) - f(x_0)}{x - x_0}$. + Allora, per $x \to x_0$, per il teorema di de l'Hopital, $f'(x_0) = \lim_{x \to x_0} f'(x)$. + \end{proof} \end{document} \ No newline at end of file diff --git a/Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.pdf b/Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.pdf index 267f2552810fb5ff188857dd33195d17fc75b310..c6ccc74a4f2fcc57ecc1655bf891cddea8b959d5 100644 GIT binary patch delta 5319 zcmah~c_37M+a^nhu`dzElC5ISEN9O2*w=^%$r59UC_7my4w5}W$dV1axFORIySye_IjWch`x?xok0rqgc>iH!`os)>eXNu=mK zOuj_Hu{R?jHuX=|=Ts_fFnasWyM9?rW0;B&IKyuVG^beCJt1$@T-uzU^&e9|npkmN zQ@1`hucNa^^R_`Q`?Db_vg1YAP7@daT^N(>?V@n~)kgHEEtn>+@44FW^1B05<=L{B zjQUGAc{@1g-Wy5oCH88b>#JIemP<{s;w~IfU9ZT|zNjWM;C3c#Cj6F6%`NYUn^C?i zNHRY?E7*y|eE>erTrwUu)bKb5Bj`0n%tmQ*d*J?;xCa2|4y<4e;lwGu*moI6HO 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Introduzione al prodotto scalare/main.tex index 1cb8fdd..0c5af5b 100644 --- a/Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.tex +++ b/Geometria 1/Prodotto scalare/2023-03-22, Introduzione al prodotto scalare/main.tex @@ -172,7 +172,7 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. \begin{definition} Si dice \textbf{radicale} di un prodotto scalare $\varphi$ lo spazio: - \[ V^\perp = \{ \vec{v} \in V \mid \varphi(\vec{v}, \vec{w}) = 0, \forall \vec{w} \in V \} \] + \[ V^\perp = \{ \vec{v} \in V \mid \varphi(\vec{v}, \vec{w}) = 0 \, \forall \vec{w} \in V \} \] \vskip 0.05in \end{definition} diff --git a/Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani/main.pdf b/Geometria 1/Prodotto scalare/2023-04-17, Introduzione ai prodotti hermitiani/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..12548e74b89b94572f7e016e54ffd989cd4328ef GIT binary patch literal 206535 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intenderà uno spazio vettoriale di dimensione + finita $n$ e per $\varphi$ un suo prodotto, hermitiano o scalare + dipendentemente dal contesto. + \end{note} + + \begin{definition} (prodotto hermitiano) Sia $\KK = \CC$. Una mappa $\varphi : V \times V \to \CC$ si dice \textbf{prodotto hermitiano} se: + + \begin{enumerate}[(i)] + \item $\varphi$ è $\CC$-lineare nel secondo argomento, ossia se $\varphi(\v, \U + \w) = \varphi(\v, \U) + \varphi(\v, \w)$ e + $\varphi(\v, a \w) = a \, \varphi(\v, \w)$, + \item $\varphi(\U, \w) = \conj{\varphi(\w, \U)}$. + \end{enumerate} + \end{definition} + + \begin{definition} (prodotto hermitiano canonico in $\CC^n$) Si definisce + \textbf{prodotto hermitiano canonico} di $\CC^n$ il prodotto $\varphi : \CC^n \times \CC^n \to \CC$ tale per cui, detti $\v = (z_1 \cdots z_n)^\top$ e $\w = (w_1 \cdots w_n)^\top$, $\varphi(\v, \w) = \sum_{i=1}^n \conj{z_i} w_i$. + \end{definition} + + \begin{remark}\nl + \li $\varphi(\U + \w, \v) = \conj{\varphi(\v, \U + \w)} = + \conj{\varphi(\v, \U) + \varphi(\v, \w)} = \conj{\varphi(\v, \U)} + \conj{\varphi(\v, \U)} = \varphi(\w, \v) + \varphi(\U, \v)$, ossia + $\varphi$ è additiva anche nel primo argomento. \\ + \li $\varphi(a \v, \w) = \conj{\varphi(\w, a \v)} = \conj{a} \conj{\varphi(\w, \v)} = \conj{a} \, \varphi(\v, \w)$. \\ + \li $\varphi(\v, \v) = \conj{\varphi(\v, \v)}$, e quindi $\varphi(\v, \v) \in \RR$. \\ + \li Sia $\v = \sum_{i=1}^n x_i \vv i$ e sia $\w = \sum_{i=1}^n y_i \vv i$, allora $\varphi(\v, \w) = \sum_{i =1}^n \sum_{j=1}^n \conj{x_i} y_i \varphi(\vv i, \vv j)$. \\ + \li $\varphi(\v, \w) = 0 \iff \varphi(\w, \v) = 0$. + \end{remark} + + \begin{proposition} + Data la forma quadratica $q : V \to \RR$ del prodotto hermitiano $\varphi$ tale che $q(\v) = \varphi(\v, \v) \in \RR$, tale + forma quadratica individua univocamente il prodotto hermitiano $\varphi$. + \end{proposition} + + \begin{proof} + Innanzitutto si osserva che: + + \[ \varphi(\v, \w) = \frac{\varphi(\v, \w) + \conj{\varphi(\v, \w)}}{2} + \frac{\varphi(\v, \w) . \conj{\varphi(\v, \w)}}{2}. \] + + \vskip 0.05in + + Si considerano allora le due identità: + + \[ q(\v + \w) - q(\v) - q(\w) = + \varphi(\v, \w) + \conj{\varphi(\w, \v)} = 2 \, \Re(\varphi(\v, \w)), \] + + \[ q(i\v + \w) - q(\v) - q(\w) = -i(\varphi(\v, \w) - \conj{\varphi(\v, \w)}) = 2 \, \imm(\varphi(\v, \w)), \] + + \vskip 0.05in + + da cui si conclude che il prodotto $\varphi$ è univocamente + determinato dalla sua forma quadratica. + \end{proof} + + \begin{definition} + Si definisce \textbf{matrice aggiunta} di $A \in M(n, \KK)$ la matrice coniugata della trasposta di $A$, ossia: + + \[ A^* = \conj{A^\top} = \conj{A}^\top. \] + \end{definition} + + %TODO: aggiungere tr(conj(A^t) B) + + \begin{definition} (matrice associata del prodotto hermitiano) Analogamente + al caso del prodotto scalare, data una base $\basis = \{\vv 1, \ldots, \vv n\}$ si definisce + come \textbf{matrice associata del prodotto hermitiano} $\varphi$ + la matrice $M_\basis(\varphi) = (\varphi(\vv i, \vv j))_{i,j = 1 \textrm{---} n}$. + \end{definition} + + \begin{remark} + Si osserva che, analogamente al caso del prodotto scalare, vale + la seguente identità: + + \[ \varphi(\v, \w) = [\v]_\basis^* M_\basis(\varphi) [\w]_\basis. \] + \end{remark} + + \begin{proposition} + (formula del cambiamento di base per i prodotto hermitiani) Siano + $\basis$, $\basis'$ due basi di $V$. Allora vale la seguente + identità: + + \[ M_{\basis'} = M_{\basis}^{\basis'}(\Idv)^* M_\basis(\varphi) M_{\basis}^{\basis'}(\Idv). \] + \end{proposition} + + \begin{proof} + Siano $\basis = \{ \vv 1, \ldots, \vv n \}$ e $\basis' = \{ \ww 1, \ldots, \ww n \}$. Allora $\varphi(\ww i, \ww j) = [\ww i]_\basis^* M_\basis(\varphi) [\ww j]_\basis = \left( M_\basis^{\basis'}(\Idv)^i \right)^* M_\basis(\varphi) M_\basis^{\basis'}(\Idv)^j = + \left(M_\basis^{\basis'}(\Idv)\right)^*_i M_\basis(\varphi) M_\basis^{\basis'}(\Idv)^j$, da cui si ricava l'identità + desiderata. + \end{proof} + + \begin{definition} (radicale di un prodotto hermitiano) + Analogamente al caso del prodotto scalare, si definisce il \textbf{radicale} del prodotto $\varphi$ come il seguente sottospazio: + + \[ V^\perp = \{ \v \in V \mid \varphi(\v, \w) = 0 \, \forall \w \in V \}. \] + \end{definition} + + \begin{proposition} + Sia $\basis$ una base di $V$ e $\varphi$ un prodotto hermitiano. Allora $V^\perp = [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$\footnote{Stavolta non è sufficiente considerare la mappa $f : V \to V^*$ tale che $f(\v) = \left[ \w \mapsto \varphi(\v, \w) \right]$, dal momento che $f$ non è lineare, bensì antilineare, ossia $f(a \v) = \conj a f(\v)$.}. + \end{proposition} + + \begin{proof} + Sia $\basis = \{ \vv 1, \ldots, \vv n \}$ e sia $\v \in V^\perp$. + Siano $a_1$, ..., $a_n \in \KK$ tali che $\v = a_1 \vv 1 + \ldots + a_n \vv n$. Allora, poiché $\v \in V$, $0 = \varphi(\vv i, \v) = + = a_1 \varphi(\vv i, \vv 1) + \ldots + a_n \varphi(\vv i, \vv n) = M_i [\v]_\basis$, da cui si ricava che $[\v]_\basis \in \Ker M_\basis(\varphi)$, e quindi che $V^\perp \subseteq [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$. \\ + + Sia ora $\v \in V$ tale che $[\v]_\basis \in \Ker M_\basis(\varphi)$. + Allora, per ogni $\w \in V$, $\varphi(\w, \v) = [\w]_\basis^* M_\basis(\varphi) [\v]_\basis = [\w]_\basis^* 0 = 0$, da cui si + conclude che $\v \in V^\perp$, e quindi che $V^\perp \supseteq [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$, da cui + $V^\perp = [\cdot]_\basis \inv (\Ker M_\basis(\varphi))$, ossia + la tesi. + \end{proof} + + \begin{remark} + Come conseguenza della proposizione appena dimostrata, valgono + le principali proprietà già viste per il prodotto scalare. \\ + + \li $\det(M_\basis(\varphi)) = 0 \iff V^\perp \neq \zerovecset \iff \varphi$ è degenere \\ + \end{remark} + + % TODO: valgono buona parte delle proprietà del prodotto scalare + + % TODO: aggiunge restrizione e complessificazione +\end{document} \ No newline at end of file diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index ffe61fd..b84921e 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -147,6 +147,7 @@ \newcommand{\v}{\vec{v}} \newcommand{\vv}[1]{\vec{v_{#1}}} \newcommand{\w}{\vec{w}} +\newcommand{\U}{\vec{u}} \newcommand{\ww}[1]{\vec{w_{#1}}} \newcommand{\mapstoby}[1]{\xmapsto{#1}}