In this example, it is more convenient to use  surd  command.  surd(a,3)  means  the cubic root in the real domain. The results are displayed in a more compact form.

Compare:

restart;

de := diff(y(x), x) = 3*abs(y(x))^(2/3);

dsolve(de);

and

restart;

de := diff(y(x), x) = 3*surd(y(x)^2, 3);

dsolve(de);

The latter result can be more simplified to  x - y(x)^(1/3) +_C1 = 0 ,  but Maple does not want to do this.

P:=Matrix([[ 0 , .5 , .5 , 0 , 0 , 0 ], [ 1/3 , 0 , 0 , 1/3 , 1/3 , 0 ], [ 1/3 , 0 , 0 , 0 , 1/3 , 1/3 ], [ 0 , 1 , 0 , 0 , 0 , 0 ], [ 0 , .5 , .5 , 0 , 0 , 0 ], [ 0 , 0 , 1 , 0 , 0 , 0 ]]):

pii:=Vector[row]([ a , b , c , d , e , f ]):

solve({seq(pii.P[i]=pii[i], i=1..6)});

                               {a = f, b = f, c = f, d = f, e = f, f = f}

If I understand the problem, the rotation around the vertical axis.

The plotting of the curve:

plot(1-cos(theta), theta = 0 .. 2*Pi, coords = polar);

The plotting of the body (for clarity, cutted a piece of the outer surface):

r := 1-cos(theta):

plot3d([r*cos(theta)*cos(phi), r*cos(theta)*sin(phi), r*sin(theta)], theta =0  .. 2*Pi, phi = Pi/2 .. 2*Pi, scaling = constrained,  axes=normal, view=[-2.7..2.7,-2.7..2.7,-1.4..1.4], orientation=[-160,-75,180]);

The calculations  the volumes of the bodies bounded by the outer and inner surfaces:

int(Pi*(r*cos(theta))^2, theta = (1/2)*Pi .. 3*Pi*(1/2));

int(Pi*(r*cos(theta))^2, theta = -(1/2)*Pi .. (1/2)*Pi);

This problem was solved only with the help of  plots[textplot]  command in standard worksheet. The numerator and denominator of the fraction  r  should not be too large (less than 10^9):

MixedNumber := proc (r::{integer, fraction})

local a, b, d1, d2;

if r::integer or abs(numer(r)) < denom(r) then

return plots:-textplot([0, .9, r], view = [-1 .. 1, -1 .. 1], font = [TIMES, ROMAN, 18], color = blue, axes = none) else

a := trunc(r); b := `if`(0 < r, r-trunc(r), trunc(r)-r); d1 := `if`(0 < a, length(a), length(a)+1); d2 := max(length(numer(b)), length(denom(b)));

plots:-textplot([[-0.034*(d1-1), 0.9, a], [0.1+0.034*(d2-1), 0.9, b]], view = [-1 .. 1, -1 .. 1], font = [TIMES, ROMAN, 18], color = blue, axes = none) end if;

end proc:

Two examples of the work of the procedure:

MixedNumber(17/6);

MixedNumber(-100001/9971);

Vector  v  depends on 6 parameters  A, B, k, t, x, y . The simple procedure  VecPic  plots the vector  v  for specified values of these parameters.

VecPic := proc (A, B, k, t, x, y)

local a, b, c, d;

uses plots;

a := -A*y*exp(-k*t)/(x^2+y^2);

b := A*x*exp(-k*t)/(x^2+y^2);

c := B*t;

d := max(abs(a), abs(b), abs(c));

arrow(<a, b, c>, view = [-d .. d, -d .. d, -d .. d], color = red, scaling = constrained, axes = normal)

end proc:

Your example for  t=0, A=1 (I took the the remaining parameters arbitrarily):

VecPic(1, 1, 1, 0, 4, -3);

If I understand correctly, only zero elements are on the diagonal, and the remaining elements are equal to  0  or  1  in any combination.

Here is the procedure that builds all such symmetric  n  by  n  matrices:

LM:=proc(n)

local L;

uses combinat; 

L:=permute([1$(n*(n-1)/2), 0$(n*(n-1)/2)], n*(n-1)/2);

[seq(Matrix(n,{seq(seq((i+1,j)=L[k][(i-1)*i/2+j], j=1..i), i=1..n-1)}, shape=symmetric), k=1..nops(L))];

end proc:

Example (the first 12 matrices of  2^10  the all ones for  n=5 ):

LM(5)[1..12][];

Example:

with(LinearAlgebra):

A := Matrix(3,4, [[1,2,3,4],[5,6,7,8],[9,0,1,2]]);

SubMatrix(A, [1,2], [2,4]);

Max:=proc()

local S;

S:=select(t->not type(t, numeric), {args});

if nops(S)=0 then return max(args) else

'procname'(op(S),max({args} minus S)) fi;

end proc:

Example:

Max(3, 1, x, 4);

                       Max(x, 4)

Carl, your way from a mathematical point of view is incorrect. If a finite set of points on the curve lies in a plane, it does not mean that the entire curve lies in this plane.

Moreover  geom3d[AreCoplanar]  command can cause errors in evident examples. Look 

with(geom3d):
AreCoplanar(point(A1,[1,0,0]), point(A2,[2,0,0]), point(A3,[3,0,0]), point(A4,[1,1,0]));

    Error, (in geom3d:-plane) the points may not be AreCollinear

For a curve given by the parametric equations with smooth functions, we can use  Student[VectorCalculus][Torsion]  command.  The quote from Wiki "A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane."

Example:

A:=<cos(phi),0,-sin(phi); 0,1,0; sin(phi),0,cos(phi)>:

V:=<cos(t),sin(t),0>:

V1:=A.V;  # Circle  V  rotated through the angle phi around the axis Oy

Student[VectorCalculus][Torsion](V1, t);

After  assign(a, c);  the following two assignments  

assign(a, d);  and   a:=d;  are different things. Compare:

assign(a, c);

a := d;  b := 2;

a, b, c;

    a := d

    b := 2

    d, 2, c

plots[polarplot](1/(theta-(1/3)*Pi), theta = 1.5 .. 8);

Edited: this is a spiral. The range for the theta should be taken such that  theta-alpha<>0

The simplest way to make the bolding (input or output, or all together) is to select the desired thing by the mouse and press  B .

Addition: if you want to make the bolding throughout the worksheet, then instead of a mouse,  first select all in edit menu.

collect(algsubs(1/(T[1]*V[2])=U, B), U);

Only in standard interface:

plots[polarplot]((1/6)*Pi, r = 0 .. 3, ordering = [angular, radial]);

More short code:

L:=select(t->`*`(op(t))<>0 and nops(convert(t, set))=9, [seq(seq(seq([x, y, z, 1+x, 2+y, -3+z,-2+x, 3+y, -1+z ], z=1..9), y=1..9), x=1..9)]);

nops(L);

map(t->{a=t[1..3], b=t[4..6], c=t[7..9]}, L);