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Added 3 more diagrams.
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lines changed

‎3d37.mp

+373
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% tex/conc/mp/3d37.mp 2018-1-10 Alan U. Kennington.
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% $Id$
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% Interpretation of exterior derivative.
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input 3dmax
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input mapmax
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% verbatimtex
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% \input akmath
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% etex
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%%%%%%%%%%%%%%%%%%%%%%%%%
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% figure 1 %
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%%%%%%%%%%%%%%%%%%%%%%%%%
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beginfig(1);
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numeric A[][]; % The current 4x3 transformation matrix.
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numeric p[][], q[][]; % Lists of 3-vectors.
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pair w[]; % Coordinate pairs on the drawing canvas.
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numeric s; % The screen scale factor.
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path pat[];
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color col[];
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penAXIS := 0.5bp;
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penLN := 0.5bp;
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penTHICK := 1.2bp;
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penPT := 3.0bp;
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penBIGPT := 3.5bp;
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penARROW := 0.5bp;
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if 0=1:
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col0 := red;
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col1 := green;
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col2 := blue;
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col3 := black;
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else:
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col0 := black;
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col1 := black;
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col2 := 0.6white;
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col3 := 0.6white;
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fi
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col8 := 0.5white;
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col9 := 0.9white;
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Z_set(p0)(55, -90, 50); % Position of viewer.
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% Z_set(q0)(0, 0, 0); % Centre of picture.
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Z_set(q0)(6, 6, 6); % Centre of picture.
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A_set_pq(A)(p0)(q0);
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% A_print(A);
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s := 2000;
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unit := 10;
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%==============================================================================
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% Coordinates for the axes.
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Z_set(p10)(0cm, 0cm, 0cm); % Origin.
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Z_set(p11)(0.9unit, 0cm, 0cm); % X axis.
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Z_set(p12)(0cm, 0.9unit, 0cm); % Y axis.
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Z_set(p13)(0cm, 0cm, 0.9unit); % Z axis.
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A_calc_w(A)(w11)(p11)(s);
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A_calc_w(A)(w12)(p12)(s);
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A_calc_w(A)(w13)(p13)(s);
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A_calc_w(A)(w10)(p10)(s);
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% Draw the axes.
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pickup pencircle scaled penAXIS;
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drawarrow w10--w11;
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drawarrow w10--w12;
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drawarrow w10--w13;
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pickup pencircle scaled penPT;
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draw w10;
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label.bot(btex $x_0$ etex, w11+(0,-2pt));
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label.lrt(btex $x_1$ etex, w12+(0,0pt));
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label.lft(btex $x_2$ etex, w13+(-2pt,0pt));
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% Nominal top-left origins for m = 0 and m = 1. Origin of m=2 diagram is (0,0).
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w100 := w10 + (-4cm,5cm);
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w120 := w10 + (-4cm,0cm);
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%==============================================================================
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% Coordinates for the hyper-rectangle.
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va := 0.10unit; % Origin of local variation.
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vb := 0.20unit;
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vc := 0.20unit;
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% Dimensions of the hyper-rectangle in cubits. (One cubit = 0.523 metres.)
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waC := 20;
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wbC := 18;
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wcC := 12;
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sC := 20; % Scale factor.
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wa := (waC/sC)*unit; % Dimensions in scaled space.
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wb := (wbC/sC)*unit;
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wc := (wcC/sC)*unit;
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Z_set(p20)(va, vb, vc); % Origin of local variation.
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Z_set(p21)(va+wa, vb, vc); % X axis.
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Z_set(p22)(va, vb+wb, vc); % Y axis.
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Z_set(p23)(va, vb, vc+wc); % Z axis.
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Z_set(p24)(va+wa, vb+wb, vc); % X-Y.
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Z_set(p25)(va, vb+wb, vc+wc); % Y-Z.
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Z_set(p26)(va+wa, vb, vc+wc); % X-Z.
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Z_set(p27)(va+wa, vb+wb, vc+wc); % X-Y-Z.
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A_calc_w(A)(w20)(p20)(s);
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A_calc_w(A)(w21)(p21)(s);
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A_calc_w(A)(w22)(p22)(s);
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A_calc_w(A)(w23)(p23)(s);
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A_calc_w(A)(w24)(p24)(s);
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A_calc_w(A)(w25)(p25)(s);
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A_calc_w(A)(w26)(p26)(s);
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A_calc_w(A)(w27)(p27)(s);
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% Faces of hyper-rectangle.
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pat1 := w20--w22--w25--w23--cycle; % O Y YZ Z
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pat2 := w20--w23--w26--w21--cycle; % O Z XZ X
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pat3 := w20--w21--w24--w22--cycle; % O X XY Y
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pat5 := w22--w25--w27--w24--cycle; % Y YZ XYZ XY
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% Fill in the faces of the hyper-rectangle.
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if 0=1:
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pickup pencircle scaled penLN;
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fill pat1 withcolor col9;
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fill pat3 withcolor col9;
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fill pat5 withcolor col9;
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fill pat1 withcolor col0;
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fill pat3 withcolor col1;
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fill pat5 withcolor col2;
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fi
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% Draw the edges of the hyper-rectangle.
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pickup pencircle scaled penLN;
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% draw w20--w21;
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% draw w20--w22;
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% draw w20--w23;
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draw w21--w24;
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draw w21--w26;
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draw w22--w24;
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draw w22--w25;
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draw w23--w25;
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draw w23--w26;
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draw w24--w27;
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draw w25--w27;
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draw w26--w27;
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% Draw the main vectors.
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pickup pencircle scaled penTHICK;
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drawarrow w20--w21;
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drawarrow w20--w22;
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drawarrow w20--w23;
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pickup pencircle scaled penBIGPT;
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draw w20;
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label.llft(btex $x$ etex, w20);
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label.bot(btex\strut $V_0$ etex, w21);
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label.urt(btex\strut $V_1$ etex, w22);
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label.ulft(btex $V_2$ etex, w23);
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%==============================================================================
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% Draw the orientations of all 6 faces of the hyper-rectangle.
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dZ := 0.05unit; % Semi-width of orientation loops.
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% The YZ plane for term 0.
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Z_set(p30)(va, vb+wb/2, vc+wc/2); % YZ.
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Z_set(p31)(va, vb+wb/2-dZ, vc+wc/2-dZ);
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Z_set(p32)(va, vb+wb/2+dZ, vc+wc/2-dZ);
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Z_set(p33)(va, vb+wb/2+dZ, vc+wc/2+dZ);
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Z_set(p34)(va, vb+wb/2-dZ, vc+wc/2+dZ);
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Z_set(p40)(va+wa, vb+wb/2, vc+wc/2); % X + YZ.
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Z_set(p41)(va+wa, vb+wb/2-dZ, vc+wc/2-dZ);
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Z_set(p42)(va+wa, vb+wb/2+dZ, vc+wc/2-dZ);
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Z_set(p43)(va+wa, vb+wb/2+dZ, vc+wc/2+dZ);
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Z_set(p44)(va+wa, vb+wb/2-dZ, vc+wc/2+dZ);
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A_calc_w(A)(w30)(p30)(s);
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A_calc_w(A)(w31)(p31)(s);
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A_calc_w(A)(w32)(p32)(s);
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A_calc_w(A)(w33)(p33)(s);
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A_calc_w(A)(w34)(p34)(s);
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A_calc_w(A)(w40)(p40)(s);
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A_calc_w(A)(w41)(p41)(s);
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A_calc_w(A)(w42)(p42)(s);
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A_calc_w(A)(w43)(p43)(s);
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A_calc_w(A)(w44)(p44)(s);
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pickup pencircle scaled penLN;
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drawarrow reverse (w34--w31) withcolor col0;
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drawarrow reverse (w33--w34) withcolor col1;
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drawarrow reverse (w32--w33) withcolor col2;
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drawarrow reverse (w31--w32) withcolor col3;
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drawarrow w41--w42 withcolor col0;
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drawarrow w42--w43 withcolor col1;
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drawarrow w43--w44 withcolor col2;
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drawarrow w44--w41 withcolor col3;
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% The XZ plane for term 1.
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Z_set(p50)(va+wa/2, vb, vc+wc/2); % XZ.
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Z_set(p51)(va+wa/2-dZ, vb, vc+wc/2-dZ);
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Z_set(p52)(va+wa/2-dZ, vb, vc+wc/2+dZ);
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Z_set(p53)(va+wa/2+dZ, vb, vc+wc/2+dZ);
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Z_set(p54)(va+wa/2+dZ, vb, vc+wc/2-dZ);
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Z_set(p60)(va+wa/2, vb+wb, vc+wc/2); % Y + XZ.
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Z_set(p61)(va+wa/2-dZ, vb+wb, vc+wc/2-dZ);
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Z_set(p62)(va+wa/2-dZ, vb+wb, vc+wc/2+dZ);
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Z_set(p63)(va+wa/2+dZ, vb+wb, vc+wc/2+dZ);
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Z_set(p64)(va+wa/2+dZ, vb+wb, vc+wc/2-dZ);
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A_calc_w(A)(w50)(p50)(s);
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A_calc_w(A)(w51)(p51)(s);
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A_calc_w(A)(w52)(p52)(s);
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A_calc_w(A)(w53)(p53)(s);
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A_calc_w(A)(w54)(p54)(s);
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A_calc_w(A)(w60)(p60)(s);
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A_calc_w(A)(w61)(p61)(s);
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A_calc_w(A)(w62)(p62)(s);
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A_calc_w(A)(w63)(p63)(s);
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A_calc_w(A)(w64)(p64)(s);
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pickup pencircle scaled penLN;
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drawarrow reverse (w54--w51) withcolor col0;
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drawarrow reverse (w53--w54) withcolor col1;
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drawarrow reverse (w52--w53) withcolor col2;
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drawarrow reverse (w51--w52) withcolor col3;
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drawarrow w61--w62 withcolor col0;
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drawarrow w62--w63 withcolor col1;
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drawarrow w63--w64 withcolor col2;
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drawarrow w64--w61 withcolor col3;
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% The XY plane for term 2.
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Z_set(p70)(va+wa/2, vb+wb/2, vc); % XY.
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Z_set(p71)(va+wa/2-dZ, vb+wb/2-dZ, vc);
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Z_set(p72)(va+wa/2+dZ, vb+wb/2-dZ, vc);
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Z_set(p73)(va+wa/2+dZ, vb+wb/2+dZ, vc);
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Z_set(p74)(va+wa/2-dZ, vb+wb/2+dZ, vc);
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Z_set(p80)(va+wa/2, vb+wb/2, vc+wc); % Z + XY.
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Z_set(p81)(va+wa/2-dZ, vb+wb/2-dZ, vc+wc);
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Z_set(p82)(va+wa/2+dZ, vb+wb/2-dZ, vc+wc);
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Z_set(p83)(va+wa/2+dZ, vb+wb/2+dZ, vc+wc);
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Z_set(p84)(va+wa/2-dZ, vb+wb/2+dZ, vc+wc);
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A_calc_w(A)(w70)(p70)(s);
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A_calc_w(A)(w71)(p71)(s);
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A_calc_w(A)(w72)(p72)(s);
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A_calc_w(A)(w73)(p73)(s);
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A_calc_w(A)(w74)(p74)(s);
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A_calc_w(A)(w80)(p80)(s);
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A_calc_w(A)(w81)(p81)(s);
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A_calc_w(A)(w82)(p82)(s);
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A_calc_w(A)(w83)(p83)(s);
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A_calc_w(A)(w84)(p84)(s);
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pickup pencircle scaled penLN;
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drawarrow reverse (w74--w71) withcolor col0;
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drawarrow reverse (w73--w74) withcolor col1;
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drawarrow reverse (w72--w73) withcolor col2;
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drawarrow reverse (w71--w72) withcolor col3;
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drawarrow w81--w82 withcolor col0;
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drawarrow w82--w83 withcolor col1;
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drawarrow w83--w84 withcolor col2;
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drawarrow w84--w81 withcolor col3;
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% Some labels.
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w90 := w40 + (0pt,-18pt);
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w91 := w40 + (0pt,-96pt);
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w92 := w60 + (3pt,13pt);
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w93 := w60 + (20pt,55pt);
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w94 := w80 + (-15pt,4pt);
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w95 := w80 + (-90pt,30pt);
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w96 := w80 + (-90pt,30pt);
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pickup pencircle scaled penARROW;
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drawarrow w91--w90 dashed evenly;
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label.bot(btex $\partial_{V_0}\omega(x)(V_1,V_2)$ etex, w91);
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drawarrow w93--w92 dashed evenly;
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label.top(btex $\partial_{V_1}\omega(x)(V_2,V_0)$ etex, w93);
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drawarrow w95--w94 dashed evenly;
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label.lft(btex $\partial_{V_2}\omega(x)(V_0,V_1)$ etex, w95);
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label.top(btex $m=2$ etex, w25+(-20pt,0pt));
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%==============================================================================
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% The case m = 0.
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dQ := 2cm;
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dL := 14pt;
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w101 := w100 + (0,0);
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w102 := w100 + (dQ,0);
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w103 := 0.5[w101,w102]+(0,dL);
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w104 := 0.5[w101,w102]+(0,-dL);
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pickup pencircle scaled penLN;
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draw w101--w102 dashed evenly withcolor col8;
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pickup pencircle scaled penPT;
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draw w101;
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draw w102;
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label.top(btex\strut $-$ etex, w101);
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label.top(btex\strut $+$ etex, w102);
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label.bot(btex\strut $x$ etex, w101);
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label.bot(btex\strut $V_0$ etex, w102);
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label.top(btex $m=0$ etex, w103);
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label.bot(btex $\partial_{V_0}\omega(x)()$ etex, w104);
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%==============================================================================
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% The case m = 1.
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w121 := w120 + (0,0);
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w122 := w120 + (dQ,0);
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w123 := w120 + (dQ,dQ);
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w124 := w120 + (0,dQ);
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w130 := 0.5[w124,w123] + (0,dL);
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w131 := 0.5[w121,w122] + (0,-dL);
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w132 := 0.5[w121,w122] + (0,-2dL);
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pickup pencircle scaled penARROW;
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drawarrow w121--w122 withcolor col0;
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drawarrow w122--w123 withcolor col1;
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drawarrow w123--w124 withcolor col2;
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drawarrow w124--w121 withcolor col3;
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pickup pencircle scaled penPT;
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draw w121;
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label.llft(btex\strut $x$ etex, w121);
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label.lrt(btex\strut $V_0$ etex, w122);
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label.ulft(btex\strut $V_1$ etex, w124);
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label.lft(btex $-$ etex, 0.5[w121,w124]);
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label.rt(btex $+$ etex, 0.5[w122,w123]);
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label.bot(btex $+$ etex, 0.5[w121,w122]);
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label.top(btex $-$ etex, 0.5[w124,w123]);
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label.top(btex $m=1$ etex, w130);
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label.bot(btex $\partial_{V_0}\omega(x)(V_1)$ etex, w131);
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label.bot(btex $\llap{${}-{}$}\partial_{V_1}\omega(x)(V_0)$ etex, w132);
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endfig;
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end

‎dgmpmax.tex

+6-2
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@@ -1,4 +1,4 @@
1-
% tex/conc/dgmpmax.tex 2017-11-24 Alan U. Kennington.
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% tex/conc/dgmpmax.tex 2018-1-13 Alan U. Kennington.
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% $Id: tex/conc/dgmpmax.tex fb0adf70be 2017-11-24 12:18:46Z Alan U. Kennington $
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% Some macros for use in the DG book and MetaPost files.
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% General mathematics things should go in akmath.tex.
@@ -104,7 +104,7 @@
104104
% Could also define \overlineleftright and \overlineleft etc.?
105105
\def\overlineright#1#2{%
106106
{\setbox0\hbox{$#2$}\hbox to\wd0{\hss$\overline{\hbox to#1{\hss$#2$}}$}}}
107-
\def\XXoline{\overlineright{9pt}{\XX}{}}
107+
\def\XXoline{\overlineright{8pt}{\XX}{}}
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109109
\let\XXcut\XXoline % Short-cut de-tagged form cross-sections.
110110

@@ -459,3 +459,7 @@
459459
\def\Slice{\textop{Slice}}
460460
\def\Subst{\textop{Subst}}
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\def\opLift{\textop{Lift}} % Not to be confused with \lift.
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% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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% Constant vector field extensions of vectors on differentiable manifolds.
465+
\def\Extn{\textop{Extn}}

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