@Kitonum 

An unpleasant fact is that the plot3d routine is not able to draw this cylinder properly
when z is given explicitely.

expr:=(x-y)^2+(y-z)^2+(z-x)^2 = 3:
z1,z2:=solve(expr,z):
plot3d([z1,z2],x=-3..3,y=-3..3,grid=[100,100], scaling=constrained);

plot3d([z1,z2],x=-3..3,y=x-sqrt(2)..x+sqrt(2),grid=[100,100], scaling=constrained);

@Bachatero 

Unfortunately this is theoretically impossible due to Richardson's_theorem.

Note that even now Maple is not useless here:

E:=-n*mu+ln(product(mu^x[i]/factorial(x[i]), i = 1 .. n)):
E4:=eval(E,n=4):
extrema( E4,{},mu,'sol'):sol;


It remains to extrapolate ...

For other curves, use Path(...) instead of Line(...). Type
?PathInt
for examples.

The integrals of P dx + Q dy ...   are line integrals. See
?LineInt

@1valdis 

As I understand (not being a specialist or working in string theory) the link refers to a projection of a higher dimensional manifold into R^3.
Before asking such a question you should provide the wanted projection, i.e. the parametric surface, preferable in Maple notation.

@acer 

Yes, I forgot about it. So, no need to return unevaluated by hand as I did.

@Bachatero 

Maple is not intended (prepared?) for such manipulations.
Just consider:

restart;
fn:=sum(x[i]^2,i=1..n);
f5:=eval(fn,n=5);

diff(fn, x[1]) assuming n::posint;
                               0
diff(f5, x[1]);
                             2 x[1]

[You can't use Maple e.g. in Functional Analysis. Here one needs pen+paper+brain for the moment].

@WilburC 

If f is a procedure (such as f:=sin) then

evalf(Int(f, x=0..1))

is incorrect because actually f has nothing to do with x.
(even if Maple could have been less strict and accept it).

Executing
evalf(Int(f, 0..1))
Maple will call f only with numeric arguments
which is not the case for
evalf(Int(f(x), x=0..1))
where f(x) is evaluated first, x being a symbol, not a numeric.
Your initial g called with a nonnumeric argument will produce an error, that's why it must be "unevaluated".
My previous version returns unevaluated when the argument is nonnumeric, so it needs not being unevaluated..

@WilburC 

Probably you have an older version of Maple.
Search for "premature evaluation" to understand the role of the quotes here.

Here is a version of g which does not need quotes.

g:=proc(a)
local y;
if type(a,numeric) then fsolve(a*y^2-sin(y),y=2)
else 'g'(a) end if;
end proc:

evalf( Int(g(x),x=1..2));

@Bendesarts 

This is certainly possible using directly components (instead of Explore which uses such components internally).
Of course using Explore is easier but a little limited in this case; I would switch to components only if the present simple solution is definitely inacceptable.

@Carl Love 
You mean of course
evalf(Int(x-> h(g(x)), 1..2));

@Bachatero 

It would not be difficult to write a procedure `combine/product`  to transform e.g.

product(a^p[i]*b[i],i=1..n);
into
a^sum(p[i],i=1..n)*product(b[i],i=1..n);
but I am not sure whether it's worth it. Note that actually the second expression is longer.

What has to do ec1 with your question?

@Markiyan Hirnyk 

restart;
Order:=15:
ec:=sin(x+y)+sin(x)-y:
ec0:=simplify(subs(x=z+Pi,ec)):
solve(series(ec0,y),y):
R:=subs(z=x-Pi,%);

   R := -x+Pi+(1/12)*(x-Pi)^3-(1/240)*(x-Pi)^5+(1/10080)*(x-Pi)^7+(17/362880)*(x-Pi)^9-(1153/159667200)*(x-Pi)^11+(13297/24908083200)*(x-Pi)^13+O((x-Pi)^15)

series( sin(x+R)+sin(x)-R, x=Pi ); #check
    O((x-Pi)^15)

@Markiyan Hirnyk 

@Carl Love 

# A simple proc, un-optimized, un-tested that
# should return the RootOfs with multiplicity >= m.
restart;
mindets:=proc(e,m)
local ss,ii,fs;

   fs := proc(u)   
   if nops([StringTools:-SearchAll(convert(u,string),ss)])   
      >=m then true else false
   fi
   end;

ss:=convert(e,string);
ii:=indets(e,RootOf);
select(fs,ii)
end:

e:= 10 + sin(RootOf(a^5-5)+RootOf(a^9-9)+RootOf(c^3-3)+sqrt(RootOf(z^3-3)));
mindets(e,2);