Kitonum

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These are replies submitted by Kitonum

@dharr  But if we use the  is  command to check for identity, we don't get the desired result:

is(sqrt(x+2*sqrt(x-1))=sqrt(x-1)+1);
is(sqrt(x+2*sqrt(x-1))=sqrt(x-1)+1) assuming x>=1 ;

                                    

When trying to calculate this sum symbolically, we get an incorrect result (returns  0 ):

restart;
sum(sum(`if`(igcd(m,n)=1, 1/(m^2*n^2), 0), m=1..infinity), n=1..infinity);

                                   0

@nm  Maple probably thinks that  is no simpler than  A
It's also worth noting that  expand  is a standard command in Maple for breaking an algebraic fraction (with a sum in the numerator) into the sum of several fractions:

expand((a+b+c)/d);
expand((a+b+c)/(d+e));

                 

@nm  I think OP himself should clarify which option he had in mind.

You can use the procedure  Partition  from my post  https://mapleprimes.com/posts/200677-Partition-Of-An-Integer-With-Restrictions . This procedure works directly with the branch and bound method. It finds all partitions of a positive integer   with various restrictions. Below is an example of how the procedure works. We find all partitions of the number 100, where the number of partition members will be from  1  to  , and the members themselves are greater than 1.

L:=Partition(100, 1..5, 2):
nops(L);
L[1..100][];
       

@Ronan Thanks for the explanation, now I understand your solution. Vote up. Just note that this solution contains a logical gap. In fact, you have proven that if such a configuration as in the original figure exists, then the desired area is 23.

In Maple, the number pi should be encoded as Pi, not pi.

exp(I*Pi);

   #   -1

In my opinion, this whole check is so obvious and is done manually in a few seconds (using periodicity of sinus)  that it seems a bit strange to involve Maple here. 

@jalal  Use  Matrix(2,2, ... )  instead of  Vector[row]

@SSMB I understood your original question as simply replacing the symbols in the f[3] expression with some expressions you specified. I am completely unfamiliar with the specific issues involved in your task. I can't help you any further.

@mmcdara  The variant with

n->Matrix(n,(i,j)->a[i,j])

is more universal, I often use it and therefore remembered it well. It is suitable when instead of  a[i, j]  stands any expression , depending on  and  j .

@jalal 

restart;
F:=proc(t)
local P1, P2;
uses plots;
P1:=plot3d([y^2*cos(alpha),y,y^2*sin(alpha)],y=1..2,alpha=Pi/2..t, color="LightGreen");
P2:=spacecurve([y^2*cos(t),y,y^2*sin(t)],y=1..2, color=red, thickness=3);
display(P1,P2);
end proc: 
P:=plots:-spacecurve([0,t,t^2], t=-3..3, color=red, thickness=3):
P1:=plot([[0,0]], x=-4..4,y=-3..3);
f:=plottools:-transform((x, y)->[x, y, 0]):
plots:-animate(F,[t], t=Pi/2..2*Pi+Pi/2, frames=90, labels=[x,y,z], background=P, paraminfo=false, axes=normal, axis[3]=[color=white],view=[-4..4,-3..3,-4..9]);

                     

 

@jalal  The following works :

restart;
F := t->plot3d([(y-1)*(y+2)*y*cos(alpha), y, (y-1)*(y+2)*y*sin(alpha)], y = -2.5 .. 0.5, alpha = (1/2)*Pi .. t, color = "LightBlue"):
P := plots:-spacecurve([0, t, (t-1)*(t+2)*t], t = -3 .. 2, color = red, thickness = 2):
plots:-animate(F, [t], t = (1/2)*Pi .. 2*Pi+(1/2)*Pi, frames = 90, labels = [x, y, z], background = P, paraminfo = false, axes = normal, view = [-4 .. 4, -3 .. 3, -4 .. 9], orientation=[-20,65]);

 

@nm  For problems with a parameter in Mathematica there is a very powerful  Reduce  command that does a full analysis of an equation or inequality:

 

I think developers should pay attention to this post, because none of the natural approaches give the desired result (code in Maple 2018.2):

solve( x^2+(y-2)^2=0, {x,y}) assuming real;  # Incorrect output
RealDomain:-solve( x^2+(y-2)^2=0, {x,y});  # NULL

                                   

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