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% (find-xournalpp "/tmp/2021-1-C3-matriz-jacobiana.pdf")
% file:///home/edrx/LATEX/2021-1-C3-matriz-jacobiana.pdf
% file:///tmp/2021-1-C3-matriz-jacobiana.pdf
% file:///tmp/pen/2021-1-C3-matriz-jacobiana.pdf
% http://angg.twu.net/LATEX/2021-1-C3-matriz-jacobiana.pdf
% (find-LATEX "2019.mk")
% (find-CN-aula-links "2021-1-C3-matriz-jacobiana" "3" "c3m211j" "c3j")
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% «.video-2» (to "video-2")
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%
% «.djvuize» (to "djvuize")
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% (c3m211ja "video-1")
% (find-ssr-links "c3m211j" "2021-1-C3-matriz-jacobiana" "kMGtZk5er9w")
% (code-eevvideo "c3m211j" "2021-1-C3-matriz-jacobiana" "kMGtZk5er9w")
% (code-eevlinksvideo "c3m211j" "2021-1-C3-matriz-jacobiana" "kMGtZk5er9w")
% (find-c3m211jvideo "0:00" "5/ago/2011")
%
% «video-2» (to ".video-2")
% (c3m211ja "video-2")
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% (code-eevvideo "c3m211j2" "2021-1-C3-matriz-jacobiana-2" "D_YKka3RG9E")
% (code-eevlinksvideo "c3m211j2" "2021-1-C3-matriz-jacobiana-2" "D_YKka3RG9E")
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% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
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%
% «title» (to ".title")
% (c3m211jp 1 "title")
% (c3m211ja "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 3 - 2021.1}
\bsk
Aula 21: a matriz jacobiana
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2021.1-C3.html}
\end{center}
\newpage
Comece relendo o trecho ao redor da
página 256 no capítulo 7 do Bortolossi...
Hoje nós vamos ver dois jeitos de visualizar
o que a matriz jacobiana quer dizer.
\bsk
Vamos trabalhar em cima de um exemplo só:
a função que leva cada número complexo $z∈\C$
em $z^2∈\C$.
\msk
\ColorRed{Veja os vídeos!}
\msk
Vídeo 1:
{\footnotesize
\url{http://angg.twu.net/eev-videos/2021-1-C3-matriz-jacobiana.mp4}
\url{https://www.youtube.com/watch?v=kMGtZk5er9w}
}
\ssk
Vídeo 2:
{\footnotesize
\url{http://angg.twu.net/eev-videos/2021-1-C3-matriz-jacobiana-2.mp4}
\url{https://www.youtube.com/watch?v=D_YKka3RG9E}
}
\newpage
Porque a jacobiana de $w=z^2$ é desse jeito?
$$\begin{array}{rcl}
w &=& z^2 \\
z &=& x+iy \;\;=\;\; (x,y) \;\;=\;\; \psm{x \\ y} \\
w &=& a+ib \;\;=\;\; (a,b) \;\;=\;\; \psm{a \\ b} \\[5pt]
\psm{a\\b} &=& w \\
&=& z^2 \\
&=& (x+iy)^2 \\
&=& x^2 + 2ixy + i^2y^2 \\
&=& x^2 + 2ixy - y^2 \\
&=& (x^2 - y^2) + i(2xy) \\
&=& (x^2 - y^2, 2xy) \;\;=\;\; \psm{x^2-y^2 \\ 2xy} \\
a &=& x^2 - y^2 \\
b &=& 2xy \\
\end{array}
$$
\newpage
Porque a jacobiana de $w=z^2$ é desse jeito? (2)
$$\begin{array}{rcl}
w_z Δz &=& \frac{d \psm{a\\b}}{d \psm{x\\y}} \pmat{Δx\\Δy} \\
&=& \pmat{a_x & a_y \\ b_x & b_y} \pmat{Δx\\Δy} \\
&=& \pmat{a_x Δx + a_y Δy \\ b_x Δx + b_y Δy} \\
&≈& \pmat{Δa \\ Δb} \\
&=& Δw \\
\end{array}
$$
Se $w_x$ fosse $\pmat{a_x & b_x \\ a_y & b_y}$ ao invés de
$\pmat{a_x & a_y \\ b_x & b_y}$ isso não daria certo.
\newpage
{\bf Duas figuras do ``Visual Complex Analysis'' do Needham}
Do capítulo 4, páginas 190 e 191...
% (find-latexscan-links "C3" "20210825_needham_fig1")
% (find-xpdf-page "~/LATEX/2021-1-C3/20210825_needham_fig1.pdf")
$$\includegraphics[height=3.0cm]{2021-1-C3/20210825_needham_fig1.pdf}$$
%
% (find-latexscan-links "C3" "20210825_needham_fig2")
% (find-xpdf-page "~/LATEX/2021-1-C3/20210825_needham_fig2.pdf")
$$\includegraphics[height=3.0cm]{2021-1-C3/20210825_needham_fig2.pdf}$$
% (find-bortolossi7page (+ -238 256) "matriz jacobiana")
% (find-books "__analysis/__analysis.el" "needham")
% (find-needhampage (+ 21 189) "Differentiation: The Amplitwist")
% (find-needhamtext (+ 21 189) "Differentiation: The Amplitwist")
% (find-fline "~/2021.1-C3/" "20210825_needham_fig1.jpg")
\newpage
% «exercicio-1» (to ".exercicio-1")
% (c3m211jp 6 "exercicio-1")
% (c3m211ja "exercicio-1")
{\bf Exercício 1.}
Desenhe um plano pros valores de $z$, à esquerda, e um plano pros
valores de $w=z^2$ à direita, como nas figuras do Needham. Desenhe no
planos dos `$z$'s os 9 valores de $z$ que têm $x∈\{0,1,2\}$ e
$y∈\{0,1,2\}$. Pra cada um desses 9 `$z$s calcule o `$w$
correspondente e desenhe ele no plano dos `$w$'s.
\newpage
% «preparacao-mini-teste» (to ".preparacao-mini-teste")
% (c3m211jp 7 "preparacao-mini-teste")
% (c3m211ja "preparacao-mini-teste")
\thispagestyle{empty}
\begin{center}
\vspace*{1.5cm}
\begin{tabular}{c}
{\bf \Large Exercícios de} \\
{\bf \Large preparação pro} \\
{\bf \Large mini-teste} \\
\end{tabular}
\end{center}
\newpage
No exercício 1 você descobriu as imagens
pela função $z \mapsto w$ de 9 {\sl pontos}. Agora você
vai descobrir as imagens de 9 {\sl retângulos}.
\msk
Sejam $α$ e $β$ dois reais positivos bem pequenos.
Vamos fazer os desenhos todos à mão, então você
pode usar $α=β=0.2$, ou $α=β=0.1$, algo assim ---
basta que $α^2$ e $β^2$ sejam ``desprezíveis'' no sentido
do Silvanus Thompson (e dos vídeos).
\msk
Em cada um dos 9 pontos você vai imaginar
um retangulinho de lados $α$ e $iβ$ ``apoiado nele'',
como o da figura do próximo slide...
\newpage
% (find-latexscan-links "C3" "20210827_rect_alpha_beta")
% (find-xpdf-page "~/LATEX/2021-1-C3/20210827_rect_alpha_beta.pdf")
$$\includegraphics[height=6.5cm]{2021-1-C3/20210827_rect_alpha_beta.pdf}$$
\newpage
...e você vai usar as derivadas pra encontrar uma
{\sl aproximação bastante razoável} pra imagem do
retangulinho apoiado em cada um dos 9 `$z_0$'s,
e vai desenhar essa aproximação apoiada no $w_0$
correspondente a aquele $z_0$.
\bsk
\bsk
O vídeo tem montes de explicações e dicas. Links:
\msk
{\scriptsize
\url{http://angg.twu.net/eev-videos/2021-1-C3-matriz-jacobiana-2.mp4}
\url{https://www.youtube.com/watch?v=D_YKka3RG9E}
}
\newpage
% «mt-dica-contas» (to ".mt-dica-contas")
% (c3m211jp 11 "mt-dica-contas")
% (c3m211ja "mt-dica-contas")
Se $γ=α+iβ$, então...
(Veja o vídeo!!!)
%
$$\scalebox{0.9}{$
\begin{array}{rcl}
w(z_0 + γ) &=& w(z_0 + (α+iβ)) \\
&=& w((x_0 + iy_0) + (α+iβ)) \\
&=& w((x_0 + α) + i(y_0 + β)) \\
&=& w(x_0 + α, y_0 + β) \\
&=& w(x_0 + α, y_0 + β) \\
&=& a(x_0 + α, y_0 + β) + ib(x_0 + α, y_0 + β) \\
&=& (a(x_0 + α, y_0 + β), b(x_0 + α, y_0 + β)) \\
&≈& (a + a_x α + a_y β, b + b_x α + b_y β) \\
&=& (a,b) + (a_x α + a_y β, b_x α + b_y β) \\
&=& (a,b) + \pmat{a_x & a_y \\ b_x & b_y} \pmat{α \\ β} \\
w(z_0 + γ) - w_0 &=& \pmat{a_x & a_y \\ b_x & b_y} \pmat{α \\ β} \\
\end{array}
$}
$$
\newpage
{\bf Exercício 2.}
Qual é a imagem pela função $z \mapsto z^2$ da figura abaixo?
Note que o tamanho dos retangulinhos vai depender
dos valores de $α$ e $β$ que você escolheu...
% (find-latexscan-links "C3" "20210827_9_retangulinhos")
% (find-xpdf-page "~/LATEX/2021-1-C3/20210827_9_retangulinhos.pdf")
$$\includegraphics[height=5cm]{2021-1-C3/20210827_9_retangulinhos.pdf}$$
%\printbibliography
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
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%
% «djvuize» (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-fline "~/2021.1-C3/")
# (find-fline "~/LATEX/2021-1-C3/")
# (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; xpdf $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 ~/2021.1-C3/
cp -fv $1.pdf ~/LATEX/2021-1-C3/
cat <<%%%
% (find-latexscan-links "C3" "$1")
%%%
}
f 20210827_9_retangulinhos
f 20210827_rect_alpha_beta
f 20210825_needham_fig1
f 20210825_needham_fig2
f 20201213_area_em_funcao_de_theta
f 20201213_area_em_funcao_de_x
f 20201213_area_fatias_pizza
% __ __ _
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%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2021-1-C3-matriz-jacobiana veryclean
make -f 2019.mk STEM=2021-1-C3-matriz-jacobiana pdf
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