|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2022-2-C2-edos-lineares.tex")
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% (defun C () (interactive) (find-LATEXsh "lualatex 2022-2-C2-edos-lineares.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2022-2-C2-edos-lineares.pdf"))
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% (defun u () (interactive) (find-latex-upload-links "2022-2-C2-edos-lineares"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2022-2-C2-edos-lineares.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2022-2-C2-edos-lineares")
% (find-pdf-page "~/LATEX/2022-2-C2-edos-lineares.pdf")
% (find-sh0 "cp -v ~/LATEX/2022-2-C2-edos-lineares.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2022-2-C2-edos-lineares.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2022-2-C2-edos-lineares.pdf")
% file:///home/edrx/LATEX/2022-2-C2-edos-lineares.pdf
% file:///tmp/2022-2-C2-edos-lineares.pdf
% file:///tmp/pen/2022-2-C2-edos-lineares.pdf
% http://angg.twu.net/LATEX/2022-2-C2-edos-lineares.pdf
% (find-LATEX "2019.mk")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Piecewise1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Pict3D1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v C2Subst1.lua C2Formulas1.lua ~/LATEX/")
% (find-CN-aula-links "2022-2-C2-edos-lineares" "2" "c2m222edols" "c2el")
% «.defs» (to "defs")
% «.title» (to "title")
% «.links» (to "links")
% «.links-dem-complicada» (to "links-dem-complicada")
% «.links-complexos» (to "links-complexos")
% «.maxima» (to "maxima")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c2m222edols" "2022-2-C2-edos-lineares")
% (code-eevvideo "c2m222edols" "2022-2-C2-edos-lineares")
% (code-eevlinksvideo "c2m222edols" "2022-2-C2-edos-lineares")
% (find-c2m222edolsvideo "0:00")
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\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
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%
% (find-dn6 "preamble6.lua" "preamble0")
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%\input diagxy % For 2D diagrams ("%D" lines)
%\xyoption{curve} % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx21} % (find-LATEX "edrx21.sty")
\input edrxaccents.tex % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex % (find-LATEX "edrx21chars.tex")
\input edrxheadfoot.tex % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex % (find-LATEX "edrxgac2.tex")
%\usepackage{emaxima} % (find-LATEX "emaxima.sty")
%
%\usepackage[backend=biber,
% style=alphabetic]{biblatex} % (find-es "tex" "biber")
%\addbibresource{catsem-slides.bib} % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
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%
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\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
%L dofile "Piecewise1.lua" -- (find-LATEX "Piecewise1.lua")
%L dofile "QVis1.lua" -- (find-LATEX "QVis1.lua")
%L dofile "Pict3D1.lua" -- (find-LATEX "Pict3D1.lua")
%L dofile "C2Formulas1.lua" -- (find-LATEX "C2Formulas1.lua")
%L Pict2e.__index.suffix = "%"
\pu
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% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\u#1{\par{\footnotesize \url{#1}}}
\def\drafturl{http://angg.twu.net/LATEX/2022-2-C2.pdf}
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\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m222edolsp 1 "title")
% (c2m222edolsa "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2022.2}
\bsk
Aula 25: EDOs lineares
com coeficientes constantes
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.2-C2.html}
\end{center}
\newpage
% «links» (to ".links")
% (c2m222edolsp 2 "links")
% (c2m222edolsa "links")
% (find-books "__analysis/__analysis.el" "trench" "2.1 Linear First Order")
% (find-books "__analysis/__analysis.el" "trench" "5.1 Homogeneous Linear")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "3 Second-Order Linear")
% (find-books "__analysis/__analysis.el" "marsden-weinstein" "12.7. Second-Order Linear")
% (find-books "__analysis/__analysis.el" "marsden-weinstein" "12.7. Second-Order Linear")
% (find-books "__analysis/__analysis.el" "thomas" "16 Second-Order")
% (c2m221dvsp 1 "title")
% (c2m221dvsa "title")
% (c2m212introp 12 "EDOs-chutar-testar")
% (c2m212introa "EDOs-chutar-testar")
% (find-angg ".emacs" "c2q192" "f''+7f+10f =")
{\bf Links:}
\scalebox{0.6}{\def\colwidth{14cm}\firstcol{
EDOLCCs no ``Diffy Qs'' do Jiri Lebl:
{\scriptsize
% (find-books "__analysis/__analysis.el" "lebl")
% (find-books "__analysis/__analysis.el" "lebl" "2.1 Second order linear")
% (find-books "__analysis/__analysis.el" "lebl" "2.2 Constant coefficient")
% (find-diffyqspage 84 "2.2 Constant coefficient second order linear ODEs")
% https://www.jirka.org/diffyqs/diffyqs.pdf#page=84
\url{https://www.jirka.org/diffyqs/diffyqs.pdf\#page=84}
}
\msk
EDOLCCs nas notas da Cristiane Hernández:
{\scriptsize
% (find-books "__analysis/__analysis.el" "hernandez")
% (find-books "__analysis/__analysis.el" "hernandez" "24 EDOs homogêneas")
% (find-books "__analysis/__analysis.el" "hernandez" "com coeficientes constantes")
% (find-hernandezpage (+ 10 219) "26 EDO lineares homogêneas de grau n com coeficientes constantes")
% http://angg.twu.net/2015.1-C2/CALCULOIIA_EAD_Versao_Final_correcao_aulas_25_a_30.pdf#page=229
\url{http://angg.twu.net/2015.1-C2/CALCULOIIA_EAD_Versao_Final_correcao_aulas_25_a_30.pdf\#page=229}
}
\msk
Questões sobre EDOLCCs nas provas de 2022.1:
{\scriptsize
% (c2m221p2p 5 "edo-2a-ordem")
% (c2m221p2a "edo-2a-ordem")
% http://angg.twu.net/LATEX/2022-1-C2-P2.pdf#page=5
\url{http://angg.twu.net/LATEX/2022-1-C2-P2.pdf\#page=5}
% (c2m221vrp 4 "questao-2")
% (c2m221vra "questao-2")
% http://angg.twu.net/LATEX/2022-1-C2-VR.pdf#page=4
\url{http://angg.twu.net/LATEX/2022-1-C2-VR.pdf\#page=4}
% (c2m221vsap 2 "questao-1")
% (c2m221vsaa "questao-1")
% http://angg.twu.net/LATEX/2022-1-C2-VSA.pdf#page=2
\url{http://angg.twu.net/LATEX/2022-1-C2-VSA.pdf\#page=2}
}
\msk
% «links-dem-complicada» (to ".links-dem-complicada")
%
``Uma demonstração complicada:''
{\scriptsize
% (c2m221dfip 5 "demonstracao-complicada")
% (c2m221dfia "demonstracao-complicada")
% http://angg.twu.net/LATEX/2022-1-C2-der-fun-inv.pdf#page=5
\url{http://angg.twu.net/LATEX/2022-1-C2-der-fun-inv.pdf\#page=5}
}
\msk
Nessas aulas eu escrevi muitas coisas no quadro, e ainda
não digitei elas. Você pode acessar as fotos dos quadros aqui:
% (find-angg ".emacs" "c2q222")
% (find-c2q222page 41 "nov16: EDOLCCs")
% (find-c2q222page 43 "nov17: EDOLCCs (2)")
{\scriptsize
% http://angg.twu.net/2022.2-C2/C2-quadros.pdf#page=41
\url{http://angg.twu.net/2022.2-C2/C2-quadros.pdf\#page=41}
}
\msk
Material de 2019.2:
{\scriptsize
% (find-angg ".emacs" "c2q192")
% (c2q192 93 "20191101 gde aula 19: f''+7f+10f = 0 via Álgebra Linear")
% (c2q192 95 "20191101 peq aula 19: f''+7f+10f = 0 via Álgebra Linear, aviso MT")
% (c2q192 98 "20191106 peq aula 20: f''+af'+bf = 0 complexo, valor inicial, polígono")
% (c2q192 101 "20191107 gde aula 20: f''+af'+bf = 0 complexo, valor inicial")
% (c2q192 103 "20191108 gde aula 21: f''+af'+bf = 0 complexo, poligono")
% http://angg.twu.net/2019.2-C2/2019.2-C2.pdf#page=93
\url{http://angg.twu.net/2019.2-C2/2019.2-C2.pdf\#page=93}
% (c2m192tudop 10 "title")
% (c2m192tudoa "title")
% http://angg.twu.net/LATEX/2019-2-C2-tudo.pdf#page=10
\url{http://angg.twu.net/LATEX/2019-2-C2-tudo.pdf\#page=10}
}
\msk
Sobre o `$·$' em $42·f$ e em $f·g$, veja os slides 15 até 21 daqui:
{\scriptsize
% (find-books "__cats/__cats.el" "bauer-dawn")
% (find-bauerdawnpage 15 "If f is linear then")
\url{http://math.andrej.com/asset/data/the-dawn-of-formalized-mathematics.pdf}
\url{http://math.andrej.com/2021/06/24/the-dawn-of-formalized-mathematics/}
\url{https://vimeo.com/567049015}
}
\msk
O Leithold define $f+g$, $f-g$ e $f·g$ na página 36 (cap.1)
e define a notação $\left. \frac{dy}{dx} \right]_{x=x_0}$ na p.145 (sec.3.1).
% (find-books "__analysis/__analysis.el" "leithold" "(+ 17 36)" "f+g, f-g, f*g")
% (find-books "__analysis/__analysis.el" "leithold" "(+ 17 145)" "at")
\msk
% «links-complexos» (to ".links-complexos")
% (c2m222edolsp 2 "links-complexos")
% (c2m222edolsa "links-complexos")
Capítulo sobre números complexos no ``GA1'' do Acker:
{\scriptsize
% (find-books "__analysis/__analysis.el" "acker")
% (find-books "__analysis/__analysis.el" "acker" "23 Números complexos")
% http://angg.twu.net/acker/acker__ga_livro1_2019.pdf#page=141
\url{http://angg.twu.net/acker/acker__ga_livro1_2019.pdf\#page=141}
% (c3m222taylorp 1 "title")
% (c3m222taylora "title")
}
}\anothercol{
}}
\newpage
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
$\begin{array}{rcl}
(f+g)(x) &=& f(x)+g(x) \\
f+g &=& λx.f(x)+g(x) \\
\\
(k·f)(x) &=& k·f(x) \\
k·f &=& λx.k·f(x) \\
\\
(f·g)(x) &=& f(x)·g(x) \\
f·g &=& λx.f(x)·g(x) \\
\\
(Df)(x) &=& \ddx f(x) \\
Df &=& λx.\ddx f(x) \\
D &=& λf.λx.\ddx f(x) \\
\\
D(g+h) &=& (λf.λx.\ddx f(x))(g+h) \\
&=& λx.\ddx (g+h)(x) \\
&=& λx.\ddx (g(x)+h(x)) \\
&=& λx.\ddx g(x) + \ddx h(x)) \\
&=& (λx.\ddx g(x)) + (λx.\ddx h(x))) \\
&=& Dg + Dh \\
\\
D(k·g) &=& (λf.λx.\ddx f(x))(k·g) \\
&=& λx.\ddx (k·g)(x) \\
&=& λx.\ddx (k·g(x)) \\
&=& λx.k·\ddx g(x) \\
&=& k·(λx.\ddx g(x)) \\
&=& k·Dg \\
\end{array}
$
}\anothercol{
$\begin{array}{rcl}
D(λx.x^2) &=& (λf.λx.\ddx f(x))(λx.x^2) \\
&=& λx.\ddx (λx.x^2)(x) \\
&=& λx.\ddx x^2 \\
&=& λx.2x \\
\\
k·f &=& λx.k·f \\
k·(λx.e^{kx}) &=& λx.k·(λx.e^{kx})(x) \\
&=& λx.k·e^{kx} \\
k·(λx.e^{kx}) &=& λx.k·e^{kx} \\
42·(λx.e^{kx}) &=& λx.42e^{42x} \\
\\
D(λx.e^{42x}) &=& (λf.λx.\ddx f(x))(λx.e^{42x}) \\
&=& λx.\ddx (λx.e^{42x})(x) \\
&=& λx.\ddx e^{42x} \\
&=& λx.42e^{42x} \\
&=& 42·(λx.e^{kx}) \\
\\
\end{array}
$
}}
\newpage
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
$\begin{array}[t]{rcl}
e^{iθ} &=& \cos θ + i\senθ \\
\sen -θ &=& -\sen θ \\
\cos -θ &=& -\cos θ \\
e^{-iθ} &=& \cos -θ + i\sen-θ \\
&=& \cos -θ - i\sen θ \\
&=& \cos θ - i\sen θ \\
e^{iθ}+e^{-iθ} &=& 2\cos θ \\
e^{iθ}-e^{-iθ} &=& 2i\sen θ \\
\cos θ & \eqnp{9}& \frac1 2 (e^{iθ} + e^{-iθ}) \\
\sen θ &\eqnp{10}& \frac1{2i}(e^{iθ} - e^{-iθ}) \\
\cos kθ &\eqnp{11}& \frac1 2 (e^{ikθ} + e^{-ikθ}) \\
\sen kθ &\eqnp{12}& \frac1{2i}(e^{ikθ} - e^{-ikθ}) \\
e^{(α+βi)θ} &=& e^{αθ} e^{βiθ} \\
&=& e^{αθ} (\cos βθ + i \sen βθ) \\
&=& e^{αθ} \cos βθ + i e^α \sen βθ \\
e^{(α-βi)θ} &=& e^{αθ} e^{-βiθ} \\
&=& e^{αθ} (\cos(-βθ) + i \sen(-βθ)) \\
&=& e^{αθ} (\cos βθ - i \sen βθ) \\
e^{(α+βi)θ} + e^{(α-βi)θ} &=& e^{αθ} (e^{βiθ} + e^{-βiθ}) \\
&=& 2 \, e^{αθ} \cos βθ \\
e^{(α+βi)θ} - e^{(α-βi)θ} &=& e^α (e^{βiθ} - e^{-βiθ}) \\
&=& 2i \, e^{αθ} \sen βθ \\
e^α \cos βθ &=& \frac12 e^{(α+βi)θ} + \frac12 e^{(α-βi)θ} \\
e^α \sen βθ &=& \frac1{2i} e^{(α+βi)θ} - \frac1{2i} e^{(α-βi)θ} \\
\end{array}
$
}\anothercol{
$\begin{array}[t]{rcl}
E &=& e^{iθ} \\
c_α &=& \cos αθ \\
s_α &=& \sen αθ \\
E^α &=& (e^{iθ})^α \\
&=& e^{iαθ} \\
&=& c_α + is_α \\
E^{-α} &=& e^{-iαθ} \\
&=& c_{-α} + is_{-α} \\
&=& c_α - is_α \\
c_α &=& \frac1 2 (E^α + E^{-α}) \\
s_α &=& \frac1{2i}(E^α - E^{-α}) \\
E^{α+iβ} &=& \\
e^{(α+βi)θ} + e^{(α-βi)θ} &=& e^α (e^{βiθ} + e^{-βiθ}) \\
&=& 2 \, e^α \cos βθ \\
e^{(α+βi)θ} - e^{(α-βi)θ} &=& e^α (e^{βiθ} - e^{-βiθ}) \\
&=& 2i \, e^α \sen βθ \\
e^α \cos βθ &=& \frac12 e^{(α+βi)θ} + \frac12 e^{(α-βi)θ} \\
e^α \sen βθ &=& \frac1{2i} e^{(α+βi)θ} - \frac1{2i} e^{(α-βi)θ} \\
\end{array}
$
}}
\bsk
% $\ga{EDOLG=}$
% Lambda em Haskell
% Notação at
% Funções são conjuntos
% (lam181p 7 "function-graphs")
% (lam181a "function-graphs")
\newpage
% «maxima» (to ".maxima")
% (c2m222edolsp 4 "maxima")
% (c2m222edolsa "maxima")
% (find-es "maxima" "EDOLCCs")
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
% ____ _ _
% | _ \(_)_ ___ _(_)_______
% | | | | \ \ / / | | | |_ / _ \
% | |_| | |\ V /| |_| | |/ / __/
% |____// | \_/ \__,_|_/___\___|
% |__/
%
% «djvuize» (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-fline "~/2022.2-C2/")
# (find-fline "~/LATEX/2022-2-C2/")
# (find-fline "~/bin/djvuize")
cd /tmp/
for i in *.jpg; do echo f $(basename $i .jpg); done
f () { rm -v $1.pdf; textcleaner -f 50 -o 5 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf; textcleaner -f 50 -o 10 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf; textcleaner -f 50 -o 20 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 15" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 30" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 45" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.5" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.25" $1.pdf; xpdf $1.pdf }
f () { cp -fv $1.png $1.pdf ~/2022.2-C2/
cp -fv $1.pdf ~/LATEX/2022-2-C2/
cat <<%%%
% (find-latexscan-links "C2" "$1")
%%%
}
f 20201213_area_em_funcao_de_theta
f 20201213_area_em_funcao_de_x
f 20201213_area_fatias_pizza
% __ __ _
% | \/ | __ _| | _____
% | |\/| |/ _` | |/ / _ \
% | | | | (_| | < __/
% |_| |_|\__,_|_|\_\___|
%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2022-2-C2-edos-lineares veryclean
make -f 2019.mk STEM=2022-2-C2-edos-lineares pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2el"
% ee-tla: "c2m222edols"
% End: