@Carl Love  I usually insert a code text and images separately. The first by copying and pasting, and the second by making a screenshot and it's processing in Paint. Of course, it takes more time, but gives better quality pictures. I use Google Chrome, and Standard Interface for pictures.

@Carl Love   you are right. The simple example shows it very well:

plot(abs(x), x=-1..1, style=point, numpoints=20):

plot(abs(x), x=-1..1, style=point, adaptive=false, numpoints=20):

plots[display](<%% | %>, scaling=constrained);

@Carl Love  Thank you for the clarification of application of this option. Can you give an example of when  numpoints  is not enough for providing the exact number of points?

@Earl  You wrote " it appears that the surface between a pair of ribs in not a portion of a sphere but a minimal surface. Is this true? "

 It is not true. In fact, the surface of this umbrella is part of a sphere, which is above the octagon between two brocken lines  -P  and  P.

Here the plotting of this surface without the ribs with the side projecting surface:

P := piecewise(-2 < x and x < -sqrt(2), sqrt(2)*(x+2)/(2-sqrt(2)), -sqrt(2) < x and x < 0, (2-sqrt(2))*x/sqrt(2)+2, 0 < x and x < sqrt(2), (sqrt(2)-2)*x/sqrt(2)+2, sqrt(2) < x and x < 2, sqrt(2)*(x-2)/(sqrt(2)-2)):

plot3d(sqrt(-x^2-y^2+9), x = -2 .. 2, y = -P .. P, color = "HotPink", style = surface, scaling = constrained, axes = normal, numpoints = 5000, filled = true, view = [-2.45 .. 2.45, -2.45 .. 2.45, 0 .. 3.45]);

@lg674   I corrected some syntax errors in your file. But the curve is still not plotted, because the function  phi(t,1,2,3,4) takes complex values. See attached file.

Plots_new.mw

@lg674  Please give the full text of the code in which the error occurs.

@matthew_spire  I have noticed that in Maple 2015 and Maple 2016  (at least on my machine under Windows 8) by default  style=surface  occurs without lines.  If you write linestyle=solid, it will be  a surface with lines.

@Carl Love   Thanks for the detailed explanation.

Also I really liked your new one-line approach.

@J4James   Step by step solution for this integration can be done by the command

Student[Calculus1][ShowSolution](Int(exp(x)*sin(x), x));

As for  infolevel  command, I never used it. See help for this.

@acer  in acer's simplification  (unfortunately manually):

simplify(convert(evalf[9](ee), rational, exact));

subsop(5 = (exp((7/10)*r*t)*x^(7/10)+1)^2, %);

Where  the non-negativity of all the variables in your "proof"  was used? It is obvious that for arbitrary variables the inequality is not true.

@Carl Love 

Matrix.mw

@Carl Love   Should be  [T]=~[C]

@marekszpak 

A := IntegerPoints1({-1/2 < 2*f*(1/53)+7*g*(1/53), 3/106 < 7*f*(1/53)-2*g*(1/53), 2*f*(1/53)+7*g*(1/53) < -37/106, 7*f*(1/53)-2*g*(1/53) < 1/2}, [f, g]);

plots[display](plot(A, style = point, color = red, symbol = solidcircle, symbolsize = 15), plots[inequal]({-1/2 < 2*f*(1/53)+7*g*(1/53), 3/106 < 7*f*(1/53)-2*g*(1/53), 2*f*(1/53)+7*g*(1/53) < -37/106, 7*f*(1/53)-2*g*(1/53) < 1/2}, f = -2 .. 4, g = -5 .. -2, color = "LightGreen"), scaling = constrained);

@John Fredsted  The idea is the same as mine, but the implementation is much easier. I just did not know about  convert,decimal  command, etc.