Differential of Surface Area


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% File name: differential-of-surface-area.tex
% Description: 
% A geometric representation of the differential of surface area is shown.
% Date of creation: October, 23rd, 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 120 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.
	% the function $z = f(x,y)$
	\tikzmath{function funcion(\x,\y) {return 1.125+0.25*sin((0.5*\x + 1.5*\y) r);};}
	\pgfmathsetmacro{\step}{pi/50.0} % step size
	\pgfmathsetmacro{\xi}{0}	% initial value for x
	\pgfmathsetmacro{\xf}{1.0*pi}	% final value for x
	\pgfmathsetmacro{\yi}{0}	% initial value for y
	\pgfmathsetmacro{\yf}{1.0*pi}	% final value for y
	% Limits of the region in the xy plane
	% Location of the differential of surface area
	% Point 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] {$R$};
	\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);
	% boundary of the solid above the region of integration
	\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);
	% Indicating distances for $\Delta x$
	\draw[dash dot dot] (\px,\py,0) -- (\px,0,0);
	\draw[dash dot dot] (\px+\dx,\py,0) -- (\px+\dx,0,0);
	\node[left] at (\px-\dx,0,0) {$dx$};
	% Indicating distances for $\Delta y$
	\draw[dash dot dot] (0,\py,0) -- (\px,\py,0);
	\draw[dash dot dot] (0,\py+\dy,0) -- (\px,\py+\dy,0);
	\node[above] at (0,\py+\dy,0) {$dy$};	
	% Differential of area $dA$
	\draw[fill=blue,opacity=0.5] (\px,\py,0) -- (\px,\py+\dy,0) -- (\px+\dx,\py+\dy,0) -- (\px+\dx,\py) -- (\px,\py,0);
	\draw[dash dot dot] (\px,\py,0) -- (\px,\py,\zdA);
	\draw[dash dot dot] (\px+\dx,\py,0) -- (\px+\dx,\py,\zdB);
	\draw[dash dot dot] (\px,\py+\dy,0) -- (\px,\py+\dy,\zdD);
	\draw[dash dot dot] (\px+\dx,\py+\dy,0) -- (\px+\dx,\py+\dy,\zdC);
	\draw[dash dot dot] (\px+\dx,\py,0) -- (\px+\dx,\py+\dy,0);
	\draw[dash dot dot] (\px,\py+\dy,0) -- (\px+\dx,\py+\dy,0);
	% The surface: 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)});
	% The graph of $z = f(x,y)$ bounding 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)});
	% Differential of surface (upon the surface of $z = f(x,y)$)
	\draw[red,fill=red,opacity=0.75] (\px,\py,\zdA) -- (\px+\dx,\py,\zdB) 
			-- (\px+\dx,\py+\dy,\zdC) -- (\px,\py+\dy,\zdD) -- (\px,\py,\zdA);
	% Nodes to indicate the location of the differentials
	\node[red,above] at (\px,\py,\zdA) {$dS$};
	\node[blue,right] at (\px,\py+\dy,0) {$dA$};

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