The procedure  P  saves from the list  L  only those pairs of entries  a  and  b  for which  abs(a - b)  equals the minimum distance  m  between the entries of the original list.

Download P.mw

To calculate sums with specific numerical (not symbolic) limits, use the  add command instead of sum :

The above also shows how easy it is to enter a matrix using 1D input instead of 2D. No palettes needed.

Download add.mw

You missed the multiplication sign. Should be

V:= (4*Pi*(d/2)^3)/3;
eval(V, d = 6.35*мм);

In arrays, indexing can start with any integer. For example:

n:=Array(0..2, [1,2,3]);
n[0];

              n := Array(0 .. 2, {0 = 1, 1 = 2, 2 = 3})
                                            1

Use the  expand  command:

Download Q20201005_2_new.mw

restart;
z:=x+I*y:
F:=evalc(abs(z+1)*abs(z-1)=1);
 plots:-implicitplot(F, x=-3..3, y=-3..3, gridrefine=3, scaling=constrained, size=[600,300]);

1. To calculate derivatives at specific points, it is better to use the differentiation operator  D  than a combination of  diff  with  subs.
2. For a better perception of what this paraboloid looks like, it is necessary to reduce the ranges on all 3 axes.
3. Calculation confirms that the vertex of this paraboloid is (0, 0, -6).

restart;
f:=(x,y)->2*x^2+3*y^2-x*y-6;
D[1](f)(1,1); # The partial derivative by x in the point (1,1)
D[2](f)(1,1); # The partial derivative by y in the point (1,1)
p1:=plot3d(f(x,y), x=-3..3, y=-3..3, style=patchcontour, view=-7..6):
p2:=plots:-spacecurve([x,1,f(x,1)],x=-3..3, color=red, thickness=3):
p3:=plots:-spacecurve([1,y,f(1,y)],y=-3..3, color=blue, thickness=3):
plots:-display(p1,p2,p3, axes=normal);
minimize(f(x,y), x=-3..3, y=-3..3, location);

Download Q20201001_new.mw
 

Download Q20201001_new.mw

Use the option  free=...  , for example:

Download MultipleTCoefficients_new.mw

In your worksheet the function  F  was not defined correctly. Here's the revised version:

Download Adomian_Polynomials_new.mw

Perhaps for your purposes it is sufficient that the index does not take on any values, but from some finite set, for example from  {1, 2, ... ,100} :

for k from 1 to 100 do
alias(F[k]=F[k](x));
od:

whattype(F[1]);
whattype(F[33]);

The easiest way to do this is with the  plots:-inequal  command:

restart; 
with(plots): 
inequal({x-y-1 <= 0, 0.5*y^2-x-3 <= 0}, x = -4 .. 6, y = -3 .. 5, color = red);

restart;
ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = 3*x(t)-2*y(t)]:
p1:=DEtools:-dfieldplot(ode,[x(t),y(t)],t=-2..2,x=-4..4, y=-4..4,arrows=SLIM,color=coral,labels=[``,``]):
p2:=plot(3*x,x=-4..4,y=-4..4,color=magenta):
p3:=plot(x,x=-4..4,y=-4..4,color=blue):
T:=plots:-textplot([[4.5,-0.2,x],[-0.2,4.5,y]], font=[times,16]):
plots:-display(p1,p2,p3,T, view=[-4.6..4.6,-4.6..4.6], size=[600,600]);

Download Help_data_points_new1.mw

In the example below, we construct 2 vertical lines (directrixes for an ellipse) in two ways:

Edit. We can also build any straight lines using the plottools:-line  command (the third way).

Download vl-2.mw