Differential of Volume Rectangular Coordinates


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% File name: differential-of-volume-rectangular-coordinates.tex
% Description: 
% A geometric representation of the differential of 
% volume in rectangular coordinates is shown.
% Date of creation: September, 25th, 2021.
% Date of last modification: October, 9th, 2022.
% Author: Efraín Soto Apolinar.
% https://www.aprendematematicas.org.mx/author/efrain-soto-apolinar/instructing-courses/
% Source: page 121 of the 
% Glosario Ilustrado de Matem\'aticas Escolares.
% https://tinyurl.com/5udm2ufy
% Terms of use:
% According to TikZ.net
% https://creativecommons.org/licenses/by-nc-sa/4.0/
% Your commitment to the terms of use is greatly appreciated.
	\tikzmath{function funcion(\x,\y) {return 2.5+0.25*sin((0.5*\x + \y) r);};}
	% Límites de mi región en el plano xy
	% Punto donde dibujo el diferencial de área
	% Punto A (\a,\c,0)
	% Coordinate axis
	\draw[thick,->] (0,0,0) -- (\xf+0.25,0,0) node [below] {$x$}; % Eje x
	\draw[thick,->] (0,0,0) -- (0,\yf+0.25,0) node [right] {$y$}; % Eje y
	\draw[thick,->] (0,0,0) -- (0,0,\h+0.5,0) node [above] {$z = f(x,y)$}; 
	% The region in the xy-plane
	\draw[white] (\a,\d,0) -- (\b,\d,0) node [black,below,sloped,midway] {$R$};
	\draw[dash dot dot] (\a,0,0) node[above left] {$a$} -- (\a,\c,0);
	\draw[dash dot dot] (\b,0,0) node[above left] {$b$} -- (\b,\c,0);
	\draw[dash dot dot] (0,\c,0) node[above right] {$c$}-- (\a,\c,0);
	\draw[dash dot dot] (0,\d,0) node[above right] {$d$}-- (\a,\d,0);
	\fill[gray!25] (\a,\c,0) -- (\b,\c,0) -- (\b,\d,0) -- (\a,\d,0) -- (\a,\c,0);
	\draw[dash dot dot] (\a,\c,0) -- (\b,\c,0) -- (\b,\d,0) -- (\a,\d,0) -- (\a,\c,0);
	% The solid (boundary)
	\draw[dash dot dot] (\a,\c,0) -- (\a,\c,\zA);
	\draw[dash dot dot] (\b,\c,0) -- (\b,\c,\zB);
	\draw[dash dot dot] (\a,\d,0) -- (\a,\d,\zD);
	\draw[dash dot dot] (\b,\d,0) -- (\b,\d,\zC);
	% Differential of area $dA$
	\draw[fill=cyan] (\px,\py,0) -- (\px,\py+\dy,0) -- (\px+\dx,\py+\dy,0) -- (\px+\dx,\py) -- (\px,\py,0);
	% Delimiting the differential of volume
	\draw[thin] (\px,\py,0) -- (\px,\py,\zdA);
	% Face parallel to the yz plane
		(\px+\dx,\py,0) -- (\px+\dx,\py,\zdB) -- (\px+\dx,\py+\dy,\zdC) 
		-- (\px+\dx,\py+\dy,0) -- (\px+\dx,\py,0);
	% Face parallel to the xz plane
		(\px,\py+\dy,0) -- (\px,\py+\dy,\zdD) -- (\px+\dx,\py+\dy,\zdC) 
		-- (\px+\dx,\py+\dy,0) -- (\px+\dx,\py+\dy,0);
	\draw[thin] (\px+\dx,\py,0) -- (\px+\dx,\py,\zdB) node[left,midway] {$dV$};
	\draw[thin] (\px,\py+\dy,0) -- (\px,\py+\dy,\zdD);
	\draw[thin] (\px+\dx,\py+\dy,0) -- (\px+\dx,\py+\dy,\zdC);
	\draw[thin] (\px+\dx,\py,0) -- (\px+\dx,\py+\dy,0);
	\draw[thin] (\px,\py+\dy,0) -- (\px+\dx,\py+\dy,0);
	\draw[fill=cyan,opacity=0.75,draw=black] (\px,\py,\zdA) -- (\px+\dx,\py,\zdB) 
			-- (\px+\dx,\py+\dy,\zdC) -- (\px,\py+\dy,\zdD) -- (\px,\py,\zdA);	
	% Coordinate axis
	\draw[thick] (\xi-0.25,0,0) -- (0,0,0); % Eje x
	\draw[thick] (0,\yi-0.25,0,0) -- (0,0,0); % Eje y
	% Graph of $z = f(x,y)$ (first quadrant)
	\foreach \x in {0,\step,...,\xf}{
		\draw[cyan,opacity=0.5] plot[domain=0:\yf,smooth,variable=\t] ({\x},{\t},{funcion(\x,\t)});
	\foreach \y in {0,\step,...,\yf}{
		\draw[cyan,opacity=0.5] plot[domain=0:\yf,smooth,variable=\t] ({\t},{\y},{funcion(\t,\y)});
	% Curves upon the graph of $z = f(x,y)$ that delimit the region of integration
	\foreach \x in {\a,\b}
		\draw[blue,thick,opacity=0.85] plot[domain=\c:\d,smooth,variable=\t] ({\x},{\t},{funcion(\x,\t)});
	\foreach \y in {\c,\d}
		\draw[blue,thick,opacity=0.85] plot[domain=\a:\b,smooth,variable=\t] ({\t},{\y},{funcion(\t,\y)});

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