From 02dce8ebd194038e062740aa0dc2196072e8d38e Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Wed, 29 Mar 2023 18:40:20 +0200 Subject: [PATCH] feat: aggiunge gli appunti del 29/03/2023 --- Fisica I/2023-03-29/main.pdf | Bin 0 -> 126891 bytes Fisica I/2023-03-29/main.tex | 45 +++++++ .../main.pdf | Bin 0 -> 190395 bytes .../main.tex | 83 ++++++++++++- .../2023-03-27/main.pdf | Bin 169159 -> 0 bytes .../2023-03-29/main.pdf | Bin 0 -> 151972 bytes .../2023-03-29/main.tex | 110 ++++++++++++++++++ 7 files changed, 236 insertions(+), 2 deletions(-) create mode 100644 Fisica I/2023-03-29/main.pdf create mode 100644 Fisica I/2023-03-29/main.tex create mode 100644 Geometria I/Teoria spettrale degli endomorfismi/2023-03-27, Proprietà e teoremi principali sul prodotto scalare/main.pdf rename Geometria I/Teoria spettrale degli endomorfismi/{2023-03-27 => 2023-03-27, Proprietà e teoremi principali sul prodotto scalare}/main.tex (51%) delete mode 100644 Geometria I/Teoria spettrale degli endomorfismi/2023-03-27/main.pdf create mode 100644 Geometria I/Teoria spettrale degli endomorfismi/2023-03-29/main.pdf create mode 100644 Geometria I/Teoria spettrale degli endomorfismi/2023-03-29/main.tex diff --git a/Fisica I/2023-03-29/main.pdf b/Fisica I/2023-03-29/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..569f92eaf46c85ee64a988501af4a89a9f537071 GIT binary patch literal 126891 zcmbrm1zeO}v^Ppgm!ycaq%_PhbW4M@bc1x4bf>h0fP@ODNF&`1Qqqldw}cY+nVHw~ zdd~Uox!-p`e!tnXpB-zj{I4~`JP-Xdsi$mQ>>y0~k@2o+Ode`ZYDZHWOd%mm4n+$G zE4LTa+*~{mYSm;`z&VRP6g&37wl%rggpiYA8Ddb_e#Ds54SkX+JH%)u3J>c^2JzI z+SgmTH#UE)`I#SiVRzoKke2&}r#n(T+Gken{s(t3X$tye;;r`aY5}fP-B9TikHg7V zohc>XBLf31dp$2xvJ0bAyCM3-{_fJcm|d-|4Q5|c7v;LGbV)9`(D9%veJtAS!3{OS z5`hCFyU!DT!dj~amENB`pA~Adj2)8W9Yz=+K|Hl(G6w-IgYP}A82Zxhv+q05XAIJh zUD@NjuH*G?^4G4i&tA!G4gOfwy;s$aE3nnn?CQXpqAF*RPWv=tVI?l>eFn|Y2ottM ziudHh83HX~1`7s#y4;J;B9cXywQ>n(l;sXmr;^bHous4*kJNqTGoBKdbtXLCYZn;S zr+doRhn^A)8QG3K(+qB+NqTaaVIcQXA!7%HGsHdVd7EeGxd_=PL1@FuxITkhMEh!4K{ z>_AYpfsUMGm&#efLF18KOv=0WI1;VHu*EVZUar!G*4=TkrSRJjO1;VNO&n3Fw%wU`S9jbauf}Qp z6vBAIpKZZIIRt%Fw$7K!s;3%;EvlJNPmvkSyiF7l~FM z`PI7GJVZ*f9zNcjAobL4(dY7&=Sh{@8Bt&PWGs|mSX|P47i39q=%1lVt19V_qo7Uw zUXp~Pa_;k;{z9m~CrY8jX+C#U38xUIfrY&KJ$WPsUZ23pDU{WYaSiEq#B(Kw=GVx4u|=;-wzn zuBlFTT~ z#y%Yh`Aw&G4PgTwn5s?UY>wG>nr`yQAkK1AN#Z)3BGoVX&2E=wN92`&tVGl5Z*lKz zbT?d!i;f3iM5fp?NS7m#)kkmq!SMm`3#{ELl`xBrjZwTu_k_VtDBbStCb#0pBqa%!Z=rZwkAT(78^9QW^Zf1h^~>X`SDVei%3hc@APV?TX`m z@Sc(C+);IPq5M38wjK3t*-A$z=WB(`gbnMDTd5diFV^Vwd>s;d!T<*n zM0sj*ApKyywvK&-ZM4ERUuKhY+f{z)%Mr#z*OO1T6|!{*f}TC)+p7!*kyU%Ig@jbm zAD@eM8;&5vbq{{FoS^rDcHwpmPIVStEm{d#x%JoKcm{5m4U(yOzUZgp zA)$B{lGPzvz#*fG(Dn{t)*CVSV=ZA3rOW(r9#{6~vdT+QIjKZ7dOhq-=G&rS+eFmb z#YjPHLsrD?l!4X8LLG45QBvmt-HjumjDf%DG~0~GUa!bm!MUgpPeCn2?C$3N-p@|{ zbrE?If8F(uxJ|EZB1Z`zJiLSB%uMwC=K|vckKH@-L|K3M?iDx~em&ME>_PF*e>Uk~ zgQNP`+%U!Y9kbTkjrh^EV^s?6y^M)%QDMlkf8zN1QS5YG+jz)H)TFAe-_wy7XSvt8 z%sib=@sZgP={CY@OHAG=6GgeWr4{d>FD5Q z;o#;<4XtF3tb59f~jG9pj{qn@IRwqfbO3z4>jbUE-xoF z|DSDq)Lc*wpd`S&)Lbz3fIcr5HNew9F>j830 zJ*DQ*u<&xDHh|V*f7pgH0_9r|%*Dmd1qceX2LCY&a;f3M=@g;IsL&=r5r@j!S0PT|!4SKR%-js0`}KeBQD;g**N&=3gL z!cT!q2jPMM~dc>(_TIXS6$dBD*9Fvd99|GnENWm2kfv$k?0@{Kf%*6$PTIC-uVEW^O=?l(3TzCIU|F={i z&>w5UZ3905#LWjB{I50PqyOxK*%-_T22O;Jz-{r*PC)bU9()BaA5?Z;5Uhn+=7eng`y8&;H-Ea5{gtu(ST<>JLolfdHlO06cR7 zJn(Y!12*Ob^HGC%V7vmH@lo^efdS3{5`eZQ&GnV9o^Q=LYt2 zal+j0fBE*ebpJQH`g`x+G~w2P&;H#9To&M-56}&S8UhO+J`hxg5I|;NoDTw(6Yv3E zz`lIYumJiHKu=JaK@d*pI6oK&N^U@UE>3RP01(Rn0l=pLMZ-K6!UI)35NOa5Ubq1u zzy>}b#sLGsS}sl?$N*}<1|W#Ip#1Q01L6IbpTYb8S$(`b01$#`#CK>z~% z*B<=(e{KPlaRP$DU;p#c|NU7QQCRc$o_|OGocHg@UyJ`8{44PRegW|KZv&^x4PYB= z3=D?a3f@BC&P?5xj-3fawbGHXy^p+rRV(Ka&rz0T7iSFrZ}+fJs2g0DKwn z0Kk6%p8}i>ur6R5z?T6_12%@WU^pG1asWf1e!;~Jcq_03@MwSlKQwiKfgk~LEEkkr z2mmy|Auy+c5&&EaFgnziVVM;IKMEQLU@$bV^8twaN65bg{!jgXYXuzX zOwdXMS|&j2DQKAU0)PM%m@r_0@bUvAU;y%f)Cfy35KbsWKm#8N8qiV}I>!e9Dm1k~ zVFd(~6+p@W(gEO#FwMe~DpVtYY_+kdsf3-dS-um{Nf0B8Zw0;CzJ zb3=WT7Yy}EUchOAmJezpDE~Zw#eq4%qu}enKr)6vZNUfd3d3PuSjhtdWB^hn0GEKR z0s8^80e^*B3R<^<0I&zBz*h1BU#iZMyN;qCohCn%&=ks z0wiD{24EouO`5=JSkS|)@pl;iw?15Jf5QNr7QBVqoCh9-e_`PtKY`MR`##JU{-O@O z=7&{vT%3TZ0LOqa_qP*4m%xwtxA*s|zl{YW{}%-wn85$%j{Nue@IL%RSQQ4Z#s1+G zraDeIw7}31R^!6m6AXg72Or#QfC~OG=3jCG0ZYvfb1N7fxM5)K2tSVxUR1--_aB^v z1r8L(;V}BoB^+}974*OM{=J6V@E>cy2*dk-Sr0Y}-|^2V>@4`mKc;{u1gNdyJ=h8! 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@@ +\documentclass[11pt]{article} +\usepackage[physics]{personal_commands} +\usepackage[italian]{babel} + +\title{\textbf{Note del corso di Fisica 1}} +\author{Gabriel Antonio Videtta} +\date{\today} + +\begin{document} + + \maketitle + + \begin{center} + \Large \textbf{Esempi di forze conservative} + \end{center} + + Un esempio notevole di forza conservativa è quello della + forza elastica $\vec f = -k \vec r$. Sia infatti $\vec f = (f_x, f_y, f_z)$. + Allora $L_{\gamma(A, B)} = \int_{\gamma(A, B)} \vec f \cdot d\vec r = + \int_{x_A}^{x_B} f_x dx + \int_{y_A}^{y_B} f_y dy + \int_{z_A}^{z_B} f_z dz = + -k (\int_{x_A}^{x_B} x dx + \int_{y_A}^{y_B} y dy + \int_{z_A}^{z_B} z dz) = + -\frac{k}{2} (\norm{B}^2 - \norm{A}^2)$, ossia non dipende dalla traiettoria + $\gamma$. Si ricava allora che $U(x) = \frac{k}{2} x ^2$, nel caso + unidimensionale. + + %TODO: recuperare lezione. + + \begin{definition} (impulso di una forza) + Si definisce \textbf{impulso di una forza} l'integrale + $\vec I(t_1, t_2) = \int_{t_1}^{t_2} \vec F(t) dt$. + \end{definition} + + Sia $\vec F = \sum_{i=1}^N \vec F_i$. Allora $\vec I(t_1, t_2) = + \sum_{i=1}^N \vec I_i(t_1, t_2)$, dove $\vec I_i$ è calcolato su $\vec F_i$. + + \begin{theorem} (dell'impulso) + Vale l'identità $\vec I(t_1, t_2) = \vec P(t_2) - \vec P(t_1) = \Delta \vec P$. + \end{theorem} + + \begin{definition} (momento di un vettore applicato) + Si definisce \textbf{momento di un vettore} $\vec v$ dal polo + $\omega$ sul punto applicato $A$ con vettore $\vec r$ il + vettore perpendicolare ad ambo i vettori $\vec r \times \vec v$. + \end{definition} +\end{document} diff --git a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-27, Proprietà e teoremi principali sul prodotto scalare/main.pdf b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-27, Proprietà e teoremi principali sul prodotto scalare/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a38f9b901f20b319a67065d2964d8880d43a0779 GIT binary patch literal 190395 zcmbrm1z43$w?9mSbVws0-LQcjG?Ge(bc1v%DN46ND+ovn0@B?fA=04INC`+tNhtN* 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I/Teoria spettrale degli endomorfismi/2023-03-27, Proprietà e teoremi principali sul prodotto scalare/main.tex index 0d52a80..517bdaf 100644 --- a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-27/main.tex +++ b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-27, Proprietà e teoremi principali sul prodotto scalare/main.tex @@ -4,7 +4,7 @@ \title{\textbf{Note del corso di Geometria 1}} \author{Gabriel Antonio Videtta} -\date{27 marzo 2023} +\date{27 e 28 marzo 2023} \begin{document} @@ -82,5 +82,84 @@ è nulla, e quindi il prodotto scalare è nullo per la proposizione precedente. \end{proof} - %TODO: aggiungere teorema di Sylvester complesso e reale. + \begin{theorem} (di Sylvester, caso complesso) + Sia $\KK$ un campo i cui elementi sono tutti quadrati di un + altro elemento del campo (e.g.~$\CC$). Allora esiste una base + ortogonale $\basis$ tale per cui: + + \[ M_\basis(\varphi) = \Matrix{I_r & \rvline & 0 \\ \hline 0 & \rvline & 0\,}. \] + \end{theorem} + + \begin{proof} + Per il teorema di Lagrange, esiste una base ortogonale $\basis'$ di $V$. + Si riordini la base in modo tale che la forma quadratica valutata nei primi elementi sia sempre diversa da zero. Allora, poiché ogni + elemento di $\KK$ è per ipotesi quadrato di un altro elemento + di $\KK$, si sostituisca $\basis'$ con una base $\basis$ tale per + cui, se $q(\vv i) = 0$, $\vv i \mapsto \vv i$, e altrimenti + $\vv i \mapsto \frac{\vv i}{\sqrt{q(\vv i)}}$. Allora $\basis'$ + è una base tale per cui la matrice associata del prodotto scalare + in tale base è proprio come desiderata nella tesi, dove $r$ è + il numero di elementi tali per cui la forma quadratica valutata + in essi sia diversa da zero. + \end{proof} + + \begin{remark} + Si possono effettuare alcune considerazioni sul teorema di Sylvester + complesso. \\ + + \li Si può immediatamente concludere che il rango è un invariante + completo per la congruenza in un campo in cui tutti gli elementi + sono quadrati, ossia che $A \cong B \iff \rg(A) = \rg(B)$: infatti + ogni matrice simmetrica rappresenta una prodotto scalare, ed è + pertanto congruente ad una matrice della forma desiderata + nell'enunciato del teorema di Sylvester complesso. Poiché il rango + è un invariante della congruenza, si ricava che $r$ nella forma + della matrice di Sylvester, rappresentando il rango, è anche + il rango di ogni sua matrice congruente. In particolare, se due + matrici simmetriche hanno stesso rango, allora sono congruenti + alla stessa matrice di Sylvester, e quindi, essendo la congruenza + una relazione di congruenza, sono congruenti a loro volta. \\ + \li Due matrici simmetriche con stesso rango, allora, non solo + sono SD-equivalenti, ma sono anche congruenti. + \li Ogni base ortogonale deve quindi avere lo stesso numero + di elementi nulli. + \end{remark} + + \begin{theorem} (di Sylvester, caso reale) Sia $\KK$ un campo ordinato + i cui elementi positivi sono tutti quadrati (e.g.~$\RR$). Allora + esiste una base ortogonale $\basis$ tale per cui: + + \[ M_\basis(\varphi) = \Matrix{I_{i_+} & \rvline & 0 & \rvline & 0 \\ \hline 0 & \rvline & -I_{i_-} & \rvline & 0 \\ \hline 0 & \rvline & 0 & \rvline & 0\cdot I_{i_0} }. \] + \end{theorem} + + \begin{proof} + Per il teorema di Lagrange, esiste una base ortogonale $\basis'$ di $V$. + Si riordini la base in modo tale che la forma quadratica valutata nei primi elementi sia strettamente positiva, che nei secondi elementi sia strettamente negativa e che negli ultimi sia nulla. Si sostituisca + $\basis'$ con una base $\basis$ tale per cui, se $q(\vv i) > 0$, + allora $\vv i \mapsto \frac{\vv i}{\sqrt{q(\vv i)}}$; se + $q(\vv i) < 0$, allora $\vv i \mapsto \frac{\vv i}{\sqrt{-q(\vv i)}}$; + altrimenti $\vv i \mapsto \vv i$. Si è allora trovata una base + la cui matrice associata del prodotto scalare è come desiderata nella + tesi, dove $i_+$ è il numero di elementi della base la cui forma + quadratica è positiva, $i_-$ il numero di elementi cui tale forma sia + negativa e $i_0$ il numeri degli elementi cui tale forma sia nulla. + \end{proof} + + \begin{definition} + Si definisce \textbf{segnatura} di un prodotto scalare + la terna $(i_+, i_-, i_0)$, come vista nella dimostrazione + del teorema di Sylvester reale. + \end{definition} + + \begin{remark} Riguardo alla segnatura e al caso reale del teorema + di Sylvester possono essere fatte alcune considerazioni. \\ + + \li La segnatura è un invariante completo per la congruenza nel caso reale. Se infatti due matrici hanno la stessa segnatura, sono + entrambe congruenti alla matrice come vista nella dimostrazione + della forma reale del teorema di Sylvester, e quindi, essendo + la congruenza una relazione di equivalenza, sono congruenti + tra loro. + + %TODO: completare spiegazione. + \end{remark} \end{document} diff --git a/Geometria I/Teoria spettrale degli endomorfismi/2023-03-27/main.pdf b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-27/main.pdf deleted file mode 100644 index 741949d28820933d6f9d8ae2bef051f67a0abad9..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 169159 zcmbrl1zeP0(*R0}bb|;I($eh05(Yrb64Pt< ztM0=61BXY=I@%7P`Eu1sftH{3P1@Toj+tf4;Y`C3@z3rjVK1#SL^i*+&ALz>id1C( z@}&wQ6*%GjGhRwM(@>n5M_(csUB`L~x9aN0r;$a)5ssvIP&+FBLB>?=fz|)vRiC%b`0X(wa_~njPI6=ImWarL^W;m#Z66pbaG6ZVJ5>ZoDiuN z`c!M#dgN9wdO{uc{DU&hAaRn?`5p$oScTokC)@UQIK0+W>(76Z3vgxDu7pwbUbHHSLlGzOFB z7BBj&L}c+t;h)pQlyZLih40ErH#B0BftU0Myu5H5JH9U%Ppm7Ee#6&!>hUD9Rcm?H 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