In eq5 and eq6 you used square brackets [F1/V1]. This means that you created a ?list (containing only one element). Replace these by common parentheses (F1/F1).

Your example doesn't have a laurent series expansion, because the function doesn't exist for x=0 or y=0.
For a function of more variables, you can use ?mtaylor:

mtaylor( sin(x^2)/cos(y), [x,y], 12 );

If you want an asymptotic expansion, you could use a nested ?series command:

 convert( series( convert(series( sin(x)/x^2/sin(y), x ), polynom), y ), polynom ) ;
 expand(%);

You can build orbit as a dynamic Vector, that is: adding as much entries as needed, by "programmer indexing", using round parentheses instead of the common square brackets. See ?Indexing.

restart;
Orbit:=proc(Map,ic,Nit)
  local orbit,z,t:
  orbit:=Vector(1):
  z:=ic:
  for t to Nit while z<>ic or t=1 do
    orbit(t):=z;
    z:=Map(z):
  end do;
  orbit(t):=z:
  return convert(orbit,list);
end proc:

F := x -> [-x[2],x[1]]:
Orbit (F,[1,0],20);

               [[1,0], [0, 1], [-1, 0], [0, -1], [1, 0]]

The equation must be entered as

eq := LengthOfRafter = sqrt(rise^2 + run^2);

Notice that names of variables must not contain spaces. Now you have to ?solve this equation for the variable rise

The output of sole or dsolve is formatted in a way that makes it easy to use in a subs command. So in your example:

ode1:=2*diff(x(t),t)+x(t)+diff(y(t),t)+2*y(t)=exp(t):
ode2:=3*diff(x(t),t)-7*x(t)+3*diff(y(t),t)+y(t)=0:
sol:=dsolve({ode1,ode2},{x(t),y(t)}):
A:=<< -10/3|-5/3>,<17/3|4/3>>:
A.subs( sol, <x(t),y(t)> );

L := [seq(i,i=1..50)]:
for i to 10 do A[i] := mul( s[j]/(s[i]-s[j]), j=remove( n->n=i, L ) ) end do:

Now A is a ?table. Perhaps you want it to be a list:

B := convert(A,list):

expand(sin(4*x)^2); 
expand( applyrule(sin(x)^2=1-cos(x)^2, %) );

(1) I suspect that you omitted some multiplicationsigns in the definition of h

(2) To simplify a ln of an exponential, Maple has to know that all variables are real. So use the simplify command from the RealDomain package

h := r -> exp(f1 + f2*r + f3*r^2 + f4*r^3 + f5*r^4 + f6*r^5);
A := 0.2:
h(A); # Because A is a float, evalf is not needed
RealDomain:-simplify(ln(h(A)));

See Andre Heck's own homepage of the book.

In that case you should use a ?piecewise function to define such a line. I give an example:

restart;
f := x -> piecewise( x<2, x/2+1, 2*x -2 ):
g := x -> x+1:
plot([f,g], -1..4 );
X := solve( f(x)=g(x) );
Intersection := [X[1],f(X[1])], [X[2],f(X[2])];

You can remove the critical points outside the domain by a select-command:

L := CriticalPoints(f(x)):
select( x -> is(x>0), L );

simplify( subs( y= -(x^2+x*y)/2, test6 ) );
                             x + y

Add the equation

  add(gamma[i], i=1..6 ) = 1

(perhaps it is also better not to use the name gamma, because it is an ?Initially known constant.)

If this doesn'n help, upload your worksheet (use the green up-arrow)

Read the article on Lagrange Multipliers. You may also use the package ?VectorCalculus to calculate gradients, etc.

It seems that the coords-option doesn't work with the sphere()-command. So make a procedure that converts the cylindrical coordinates to cartesian:

  C := (r,theta,zeta) -> [r*cos(theta),r*sin(theta),zeta]: 

  display(sphere(C(2,Pi/2,1),1));