% (find-LATEX "2023-1-C4-P2.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-1-C4-P2.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-1-C4-P2.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023-1-C4-P2.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-1-C4-P2.pdf"))
% (defun e () (interactive) (find-LATEX "2023-1-C4-P2.tex"))
% (defun o () (interactive) (find-LATEX "2023-1-C4-P2.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-1-C4-P2"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-1-C4-P2.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2023-1-C4-P2")
% (find-pdf-page   "~/LATEX/2023-1-C4-P2.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023-1-C4-P2.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023-1-C4-P2.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2023-1-C4-P2.pdf")
%   file:///home/edrx/LATEX/2023-1-C4-P2.pdf
%               file:///tmp/2023-1-C4-P2.pdf
%           file:///tmp/pen/2023-1-C4-P2.pdf
%  http://anggtwu.net/LATEX/2023-1-C4-P2.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise1")
% (find-Deps1-cps   "Caepro5 Piecewise1")
% (find-Deps1-anggs "Caepro5 Piecewise1")
% (find-MM-aula-links "2023-1-C4-P2" "C4" "c4m231p2" "c4p2")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.defs-caepro»	(to "defs-caepro")
% «.defs-pict2e»	(to "defs-pict2e")
% «.title»		(to "title")
% «.formulas»		(to "formulas")
% «.questao-1»		(to "questao-1")
% «.links»		(to "links")
% «.stewart»		(to "stewart")
%
% «.djvuize»		(to "djvuize")



% <videos>
% Video (not yet):
% (find-ssr-links     "c4m231p2" "2023-1-C4-P2")
% (code-eevvideo      "c4m231p2" "2023-1-C4-P2")
% (code-eevlinksvideo "c4m231p2" "2023-1-C4-P2")
% (find-c4m231p2video "0:00")

\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
\usepackage{colorweb}                  % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\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")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
\begin{document}

% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

\def\drafturl{http://anggtwu.net/LATEX/2023-1-C4.pdf}
\def\drafturl{http://anggtwu.net/2023.1-C4.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2023-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2023-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

\def\div{\operatorname{div}}

% «defs-T-and-B»  (to ".defs-T-and-B")
\long\def\ColorOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorOrange{\bf(#1 pts)}}

% «defs-caepro»  (to ".defs-caepro")
%L dofile "Caepro5.lua"              -- (find-angg "LUA/Caepro5.lua" "LaTeX")
\def\Caurl   #1{\expr{Caurl("#1")}}
\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca      #1{\Cahref{#1}{#1}}

% «defs-pict2e»  (to ".defs-pict2e")
%L V = nil                           -- (find-angg "LUA/Pict2e1.lua" "MiniV")
%L dofile "Piecewise1.lua"           -- (find-LATEX "Piecewise1.lua")
%L Pict2e.__index.suffix = "%"
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt

\pu



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c4m231p2p 1 "title")
% (c4m231p2a   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 4 - 2023.1}

\bsk

Segunda prova (P2)

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2023.1-C4.html}

\end{center}

\newpage

% «formulas»  (to ".formulas")



\scalebox{0.5}{\def\colwidth{12cm}\firstcol{

% «questao-1»  (to ".questao-1")
% (c4m231p2p 2 "questao-1")
% (c4m231p2a   "questao-1")

{\bf Questão 1 (e única)}

\T(Total: 10.0 pts)

\msk

O objetivo desta questão é mostrar que esta igualdade
%
$$
  \int\!\!\!\!\int_{∂B} 𝐛F·𝐛n \, dS
    \;=\; \int\!\!\!\!\int\!\!\!\!\int_B \div\,𝐛F(x,y,z)\,dV
$$
%
é verdadeira quando $B$ é a esfera de raio 2 centrada na origem e:
%
$$𝐛F(x,y,z) \;=\; (x^3+y^3)𝐛i + (y^3+z^3)𝐛j + (z^3+x^3)𝐛k
$$

a) \B(1.0 pts) Calcule $\div F$.

\ssk

b) \B(4.5 pts) Calcule o lado esquerdo da igualdade.

\ssk

c) \B(4.5 pts) Calcule o lado direito da igualdade.


% \par \Ca{StewPtCap16p70} \standout{Exercício 8}

}\anothercol{

{\bf Algumas fórmulas:}

$$\begin{array}{rcll}
  % dA &=& dx\,dy, \quad \text{ou:} \\    % StewPtCap15p8
  (x,y,z) &=&  (ρ \senϕ \cosθ,    \\    % StewPtCap15p59
           && \;\,ρ \senϕ \senθ,  \\
           && \;\,ρ \cosϕ)        & (p.927) \\
  dV &=& dx\,dy\,dz               \\
     &=& ρ^2 \senϕ\,dρ\,dϕ\,dθ    & (p.929) \\    % StewPtCap15p61
  \\[-5pt]
  % dA &=& du\,dv                   \\
  \div 𝐛F &=& ∇·𝐛F = (\frac{∂}{∂x}, \frac{∂}{∂y}, \frac{∂}{∂z})·𝐛F & (p.979) \\   % StewPtCap16p37
  \int\!\!\!\int_S f(x,y,z) \,dS &=&
    \int\!\!\!\int_D f(𝐛r(u,v))\,|𝐛r_u × 𝐛r_v| \,dA & (p.994) \\  % StewPtCap16p52
    \\[-10pt]
  𝐛n &=& \D \frac{𝐛r_u × 𝐛r_v}{|𝐛r_u × 𝐛r_v|} & (p.997) \\  % StewPtCap16p55
    \\[-10pt]
  \D \frac{𝐛r_ϕ × 𝐛r_θ}{|𝐛r_ϕ × 𝐛r_θ|} &=& \D \frac1a \; 𝐛r(ϕ,θ) & (p.998) \\  % StewPtCap16p56
    \\[-10pt]
  \int\!\!\!\int_S 𝐛F·d𝐛S
    &=& \int\!\!\!\int_S 𝐛F·𝐛n \, dS          & (p.998) \\  % StewPtCap16p56
    &=& \int\!\!\!\int_S 𝐛F·(𝐛r_u×𝐛r_v) \, dA & (p.999) \\  % StewPtCap16p57
  \int\!\!\!\int_S 𝐛F·𝐛n \, dS
    &=& \int\!\!\!\int\!\!\!\int_E \div\,𝐛F(x,y,z)\,dV & (p.1008) \\ % StewPtCap16p66
  \end{array}
$$

\bsk

Algumas das fórmulas acima só fazem sentido no contexto certo. Você
vai receber cópias de algumas páginas do Stewart (7ª ed) em português
pra consulta; ``(p.42)'' quer dizer que aquela fórmula aparece na
página 42.

}\anothercol{
}}



% \par \Ca{StewPtCap15p8} Definição 5: $dA$
% \par \Ca{StewPtCap16p41} 16.6 Superfícies parametrizadas e suas áreas
% \par \Ca{StewPtCap16p43} Exemplo 4: $𝐛r(ϕ,θ)=\ldots$
% \par \Ca{StewPtCap16p47} Definição 6: $𝐛r(u,v)=\ldots$
% %\par \Ca{StewPtCap16p47} (Exemplo 10)
% \par \Ca{StewPtCap16p51} 16.7 Integrais de superfície
% \par \Ca{StewPtCap16p52} dS =
% \par \Ca{StewPtCap16p51} Superfícies parametrizadas: $𝐛r(u,v)=\ldots$
% \par \Ca{StewPtCap16p55} Figura 7: as duas orientações de uma superfície orientável
% \par \Ca{StewPtCap16p55} $𝐛r(ϕ,θ)=\ldots$
% \par \Ca{StewPtCap16p56} Definição 8: $\int\!\!\!\int_S 𝐛F·d𝐛S = \int\!\!\!\int_S 𝐛F·𝐛n\,dS$
% \par \Ca{StewPtCap16p57} \standout{Exemplo 4}
% \par \Ca{StewPtCap16p66} 16.9 O teorema do divergente
% \par \Ca{StewPtCap16p67} \standout{Exemplo 1}
% \par \Ca{StewPtCap16p69} \standout{Exercício 3}
% \par \Ca{StewPtCap16p70} \standout{Exercício 8}


\newpage

% «links»  (to ".links")

\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}

% «stewart»  (to ".stewart")

 (eepitch-lua51)
 (eepitch-kill)
 (eepitch-lua51)
ps = "927,929,979,994,997,998,999,1008"
for p in ps:gmatch("%d+") do
  printf("(find-stewart72ptpage (+ -489 %s))\n", p)
end
f = function (s) return s-489 end
= (ps:gsub("%d+", f))

-- file:///home/edrx/books/__analysis/stewart__calculo_7a_ed_vol_2.pdf
-- 438,440,490,505,508,509,510,519


%  ____  _             _         
% |  _ \(_)_   ___   _(_)_______ 
% | | | | \ \ / / | | | |_  / _ \
% | |_| | |\ V /| |_| | |/ /  __/
% |____// | \_/  \__,_|_/___\___|
%     |__/                       
%
% «djvuize»  (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-fline "~/2023.1-C4/")
# (find-fline "~/LATEX/2023-1-C4/")
# (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       ~/2023.1-C4/
       cp -fv        $1.pdf ~/LATEX/2023-1-C4/
       cat <<%%%
% (find-latexscan-links "C4" "$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=2023-1-C4-P2 veryclean
make -f 2019.mk STEM=2023-1-C4-P2 pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c4p2"
% ee-tla: "c4m231p2"
% End:
