An easy way to express the surface is in cylindrical coordinates. Thus,

plot3d(1/z, theta=0..2*Pi, z = 1..20, coords=cylindrical);

should do the trick.

It cannot. A 1x1 matrix (or Matrix) is not a scalar. Consider the product of three vectors/covectors: (mx1)(1xn)(nx1). You want to rearrange this as (1xn)(nx1)(mx1). However, if that were done, then associativity wouldn't hold, that is, (nx1)(mx1) is undefined.

Use LinearAlgebra:-Equal to test equality of Matrices (with expressions as elements). For more general testing, you can use verify (with appropriate verification argument).

with(LinearAlgebra):
A := <<1,2>>:
B := Copy(A):
Equal(A,B);
true
A := <<1,2>>:
B := <<1.001, 1.99>>:
verify(A,B,'Matrix'(float(5,digits=3)));
true
verify(A,B,'Matrix'(float(5,digits=4)));
false

Yes. Explain what it is you want to do.

Your comments are basically correct. The statement
(x -> x.x)(procname(x,n/2))
passes the argument procname(x,n/2) to the anonymous procedure x -> x.x. It is equivalent to doing
tmp := procname(x,n/2);
tmp.tmp
but avoids the local variable tmp. The reason that . was used rather than * is to allow this to work with Matrices:
Pow(<<1|2>,<3|4>>,-2);
[11/2 -5/2]
[ ]
[-15/4 7/4 ]

Here's one approach:

> M := Matrix([[x,0],[0,y],[y,0],[0,x]]);
[x 0]
[ ]
[0 y]
M := [ ]
[y 0]
[ ]
[0 x]
> DM := map(v -> `if`(v=0,0,rcurry(diff,v)),M);
[() -> diff(args, x) 0 ]
[ ]
[ 0 () -> diff(args, y)]
DM := [ ]
[() -> diff(args, y) 0 ]
[ ]
[ 0 () -> diff(args, x)]
> map(apply,DM,x^2+y);
[2 x 0 ]
[ ]
[ 0 1 ]
[ ]
[ 1 0 ]
[ ]
[ 0 2 x]

This doesn't solve the problem of applying it to a Matrix.

As Roman suggests, create a procedure rather than a polynomial. However, if you have already created the polynomial, you then need to convert it to a procedure. The procedure unapply is useful here.

P := a*x1^2 + 3*x2^2 + b:
f := unapply(P, [x1,x2]):
f(z1,z2);
a*z1^2+3*z2^2+b

Note that your example use of eval is incorrect. Proper usage is eval(P, [x=1,y=2]).

> LengthU := norm(u,2);
2

Unlike many other elements, Maple does not consider rtables (the low-level struture that includes Vectors) equal even if they have the same elements. The LinearAlgebra command Equal can be used to determine element by element (or just structure) equality of two Vectors. One way to do what you want is to convert the set to a list, then use ListTools:MakeUnique, passing LinearAlgebra:-Equal as the optional comparison procedure. Thus
lst := convert(s, list):
lst := ListTools:-MakeUnique(lst,1,Equal):
s := convert(lst, set);
An interesting observation is that if you had done
s := {UnitVector(1,2), UnitVector(1,2)}:
s would have been assigned a singleton. The reason for that is subtle, Maple applies its set simplification to the input before UnitVector is called, so at that point the set consistents of two identical function calls. This simplification can result in incorrect results if the functions return different values.

Try spacecurve([2*t,0,0],t=0..1)

You must have used the command with(inttrans) somewhere, otherwise Maple wouldn't have plotted the first y1. However, I have no problem with the second. What version of Maple are you using? Note that y1 is expressed in terms of the Heaviside function, which may have caused problems for some versions of Maple (my memory is fuzzy).

A mathematical, rather than programmatical, approach is the simpler

SplitPolynomialEvenOdd := proc(poly::polynom(anything,v), v)
description "return the even and odd parts of a polynomial in v";
local p,m;
p := collect(poly,v);
m := subs(v=-v,p);
return ((p+m)/2, (p-m)/2);
end proc:

Here's one approach.

assume(p::positive,l::positive,a::positive,q::positive);
eqn :=-p*(l^2-a^2)^(3/2)+a^3*q;
2 2 (3/2) 3
eqn := -p~ (l~ - a~ ) + a~ q~
eqn2 := isolate(eqn,a^3);
2 2 3/2
3 p~ (l~ - a~ )
eqn2 := a~ = -----------------
q~
map(x -> x^(2/3), %);
/ 2 2 3/2\(2/3)
3 (2/3) |p~ (l~ - a~ ) |
(a~ ) = |-----------------|
\ q~ /
combine(%,power);
/ 2 2 3/2\(2/3)
2 |p~ (l~ - a~ ) |
a~ = |-----------------|
\ q~ /
# At this point it is tricky to get Maple to distribute
# the power. Easier just to do it by hand:
eqn3 := a^2 = (p/q)^(2/3)*(l^2-a^2);
2 / p~ \(2/3) 2 2
eqn3 := a~ = |----| (l~ - a~ )
\ q~ /
sol := solve(eqn3,a):
sol[1];
// / p~ \(2/3)\ / p~ \(2/3)\1/2
||1 + |----| | |----| | l~
\\ \ q~ / / \ q~ / /
-------------------------------------
/ p~ \(2/3)
1 + |----|
\ q~ /

Again, distribute the power by hand. Is there a command that does that? It's easy to write one, though it won't be rigorous. You could try using the RealDomain package; that nearly solved the problem in one go but I couldn't figure out a clever way to factor out the common terms in the numerator and denominator.

You write "maxima", meaning more than one maximum. What do expect to get for the given expression, assuming the chosen component is one? That is, do you want to get the indices 1 and 2, corresponding to the two components whose first element is 8? If that is what you want, then the following will return the list of indices:
maxima_indices := proc(indx::posint, lst::list)
local elem,i,val;
val := max(seq(op(indx,elem),elem in lst));
return [seq](`if`(op([i,indx],lst)=val,i,NULL)
,i = 1..nops(lst))
end proc:
H1 := [[8,9,11],[8,10,11],[5,10,6]]:
map(maxima_indices,[1,2,3],H1);
[[1, 2], [2, 3], [1, 2]]