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3d22.mp
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3d22.mp
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% tex/conc/mp/3d22.mp 2012-5-7 Alan U. Kennington.
% $Id: tex/conc/mp/3d22.mp defbd13703 2012-05-07 12:05:29Z Alan U. Kennington $
% 3d graphic: Stokes theorem for a cube.
input 3dmax.mp
verbatimtex
\input akmath
etex
%%%%%%%%%%%%%%%%%%%%%%%%%
% figure 1 %
%%%%%%%%%%%%%%%%%%%%%%%%%
beginfig(1);
numeric A[][]; % The current 4x3 transformation matrix.
numeric p[][], q[][]; % Lists of 3-vectors.
pair w[]; % Coordinate pairs on the drawing canvas.
numeric s; % The screen scale factor.
path pat[];
color col[];
penAXIS := 0.5bp;
penLN := 0.5bp;
penTHICK := 1.2bp;
penINT := 0.6bp;
penINTPT := 1.8bp;
Z_set(p0)(60, -100, 20); % Position of viewer.
Z_set(q0)(0, 0, 0); % Centre of picture.
A_set_pq(A)(p0)(q0);
% A_print(A);
s := 2000;
unit := 10;
pickup pencircle scaled penLN;
% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Z_set(p10)(0cm, 0cm, 0cm); % Origin.
Z_set(p11)(unit, 0cm, 0cm); % X axis.
Z_set(p12)(0cm, unit, 0cm); % Y axis.
Z_set(p13)(0cm, 0cm, 0.9unit); % Z axis.
A_calc_w(A)(w11)(p11)(s);
A_calc_w(A)(w12)(p12)(s);
A_calc_w(A)(w13)(p13)(s);
A_calc_w(A)(w10)(p10)(s);
% showvariable w;
drawarrow w10--w11;
drawarrow w10--w12;
drawarrow w10--w13;
label.rt(btex $x_1$ etex, w11);
label.rt(btex $x_2$ etex, w12);
label.lft(btex $x_3$ etex, w13+(-2pt,0));
% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
va := 0.29unit; % Origin of local variation.
vb := 0.29unit;
vc := 0.37unit;
wa := 0.3unit; % Axes for local variation.
wb := 0.3unit;
wc := 0.3unit;
Z_set(p20)(va, vb, vc); % Origin of local variation.
Z_set(p21)(va+wa, vb, vc); % X axis.
Z_set(p22)(va, vb+wb, vc); % Y axis.
Z_set(p23)(va, vb, vc+wc); % Z axis.
Z_set(p24)(va+wa, vb+wb, vc); % X-Y.
Z_set(p25)(va, vb+wb, vc+wc); % Y-Z.
Z_set(p26)(va+wa, vb, vc+wc); % X-Z.
Z_set(p27)(va+wa, vb+wb, vc+wc); % X-Y-Z.
A_calc_w(A)(w20)(p20)(s);
A_calc_w(A)(w21)(p21)(s);
A_calc_w(A)(w22)(p22)(s);
A_calc_w(A)(w23)(p23)(s);
A_calc_w(A)(w24)(p24)(s);
A_calc_w(A)(w25)(p25)(s);
A_calc_w(A)(w26)(p26)(s);
A_calc_w(A)(w27)(p27)(s);
% The integration planes.
pat1 := w20--w22--w25--w23--cycle;
pat2 := w21--w24--w27--w26--cycle;
col1 := 0.9white;
col2 := 0.4white;
fill pat1 withcolor col1;
fill pat2 withcolor col1;
% Some linear integration paths.
sc := 0.8;
for ii=0 upto 5:
for jj=0 upto 1:
pb := 0.1*(2+jj*5);
pc := 0.1*(2.5+ii);
Z_set(p30)(va, vb+wb*pb, vc+wc*pc); % Left.
Z_set(p31)(va+wa, vb+wb*pb, vc+wc*pc); % Right.
A_calc_w(A)(w30)(p30)(s);
A_calc_w(A)(w31)(p31)(s);
pickup pencircle scaled penINT;
% draw w30--w31 dashed (evenly scaled sc) withcolor col2;
draw w30--w31 withcolor col2;
pickup pencircle scaled penINTPT;
draw w30;
draw w31;
endfor
endfor
% The integration cube.
pickup pencircle scaled penLN;
drawarrow w20--w21;
pickup pencircle scaled penTHICK;
drawarrow w20--w22;
drawarrow w20--w23;
drawarrow w21--w24;
drawarrow w21--w26;
pickup pencircle scaled penLN;
draw w22--w24;
draw w22--w25;
draw w23--w25;
draw w23--w26;
draw w24--w27;
draw w25--w27;
draw w26--w27;
label.llft(btex $e_1$ etex, w21);
label.lrt(btex $e_2$ etex, w24+(-1.5pt,0));
label.lrt(btex $e_3$ etex, w26+(1.5pt,0.7pt));
label.lrt(btex $e_2$ etex, w22+(-2.5pt,0.4pt));
label.llft(btex $e_3$ etex, w23+(-2pt,0));
% Some labels.
pickup pencircle scaled penLN;
w40 := 0.25[w30,w31];
w41 := w40 + (0.8cm,1.25cm);
label.rt(
btex $\displaystyle\int_{a_1}^{b_1}\!\! \partial_1 a_{23}(x)\,dx^1$ etex,
w41+(-2.5pt,0));
drawarrow w41--w40+(0.6pt,0.6pt);
endfig;
end