### My Favourite Maple Visualizati...

With Maple, you can create amazing visualizations that go far beyond the standard mathematical plots that you might typically expect (I wince every time I see yet another sine curve).

At your fingertips, you have

• plotting primitives that can be assembled in new and novel ways
• precise control over coloring (yay for ColorTools) and placement
• an interactive coding environment with inline plots, giving you quick visual feedback over aesthetic changes
• and a comprehensive mathematical programming language to glue everything together

Here, I thought I'd share a few of the visualizations I've really enjoyed creating over the last few years (and I'd like to emphasize 'enjoy' - doing this stuff is fun!)

Let me know if you want any of the worksheets.

Psychrometric chart with historical weather data for Waterloo, Ontario.

Ternary plot of the color of gold-silver-copper alloys

Spectrogram of a violin note played with vibrato

Colored zoom of the Mandelbrot set

Reporting dashboard for an Organic Rankine Cycle

Temperature-entropy plot of an ideal Rankine Cycle

Quaternion fractal

Historical sunpot data

Earthquake data

African literacy rates

### Double integral over...

This post is the answer to this question.

The procedure named  IntOverDomain  finds a double integral over an arbitrary domain bounded by a non-selfintersecting piecewise smooth curve. The code of the procedure uses the well-known Green's theorem.

Each section in the border should be specified by a list in the following formats :
1. If a section is given parametrically, then  [[f(t), g(t)], t=t1..t2]
2. If several consecutive sections of the border or the entire border is a broken line, then it is sufficient to set vertices of this broken line  [ [x1,y1], [x2,y2], .., [xn,yn] ] (for the entire border should be  [xn,yn]=[x1,y1] ).

Required parameters of the procedure:  f  is an expression in variables  x  and  y , L  is the list of all the sections. The sublists of the list  L  must follow in the positive direction (counterclockwise).

The code of the procedure:

restart;
IntOverDomain := proc(f, L)
local n, i, j, m, yk, yb, xk, xb, Q, p, P, var;
n:=nops(L);
Q:=int(f,x);
for i from 1 to n do
if type(L[i], listlist(algebraic)) then
m:=nops(L[i]);
for j from 1 to m-1 do
yk:=L[i,j+1,2]-L[i,j,2]; yb:=L[i,j,2];
xk:=L[i,j+1,1]-L[i,j,1]; xb:=L[i,j,1];
p[j]:=int(eval(Q*yk,[y=yk*t+yb,x=xk*t+xb]),t=0..1);
od;
var := lhs(L[i, 2]);
P[i]:=int(eval(Q*diff(L[i,1,2],var),[x=L[i,1,1],y=L[i,1,2]]),L[i,2]) fi;
od;
add(P[i], i = 1 .. n);
end proc:

Examples of use.

1. In the first example, we integrate over a quadrilateral:

with(plottools): with(plots):
f:=x^2+y^2:
display(polygon([[0,0],[3,0],[0,3],[1,1]], color="LightBlue"));
# Visualization of the domain of integration
IntOverDomain(x^2+y^2, [[[0,0],[3,0],[0,3],[1,1],[0,0]]]);  # The value of integral

2. In the second example, some sections of the boundary of the domain are curved lines:

display(inequal({{y<=sqrt(x),y>=sin(Pi*x/3)/2,y<=3-x}, {y>=-2*x+3,y>=sqrt(x),y<=3-x}}, x=0..3,y=0..3, color="LightGreen", nolines), plot([[t,sqrt(t),t=0..1],[t,-2*t+3,t=0..1],[t,3-t,t=0..3],[t,sin(Pi*t/3)/2,t=0..3]], color=black, thickness=2));
f:=x^2+y^2: L:=[[[t,sin(Pi*t/3)/2],t=0..3],[[3,0],[0,3],[1,1]], [[t,sqrt(t)],t=1..0]]:
IntOverDomain(f, L);

3. If  f=1  then the procedure returns the area of the domain:

IntOverDomain(1, L);  # The area of the above domain
evalf(%);

IntOverDomain.mw

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