@Carmelo I'm sorry, but I don't understand your English "With eval does not made." Perhaps you should show the relevant output from Maple where something hasn't been "made".

Your question about extraction of results leads me to believe that you believe that dsolve has already computed the results and they're sitting is memory somewhere and you merely need to "extract" them. That's not true; the results aren't computed until you call num_n.

@vv Thank you for the example. It's certainly simple.

@vv Can you give me a simple example of an improper integral that becomes proper after a change of variables? one where after the change the integrand is entire?                                  

The original vector field is not continuous because of the 1/y term. In light of this I have two questions---one mathematical and one about Maple. The math question is What does the flux mean in this context? How does one compute it? Can the divergence theorem be used? The Maple question is What does the integral returned by Flux mean? That integral is proper: The integrand is continuous for all phi and theta, and the limits of integration are finite. That's not believable to me.

@mriedel Yes, by "wrong" I meant that there's a bug in the original summation, not that you were wrong to try to do it!

The initial summation is wrong. There's no need to use series, large values of j, or even a symbolic z to see that. Make z=2 from the start, and evaluate at j=0:

S:= eval(sum(product(k+4-m, m=0..k-1)*2^k, k=0..j), j= 0);

Simply that the result isn't an integer shows that the summation is wrong.

@Axel Vogt Escaped local names are very much like module exports: They can be accessed from outside the code that generated them, and they are distinct from globals of the same name. Here's an example. The procedure below returns a as an escaped local, and we can see that it's different from the global a.

proc() local a; a end proc();

     a

% - a;

     a - a

Unlike a module export, there's no easy way to refer to this a. It can be done, but it's messy:

%% - indets(%%)[];

     0

In the integral that's returned by Flux, the theta and phi are escaped local names. If Acer hadn't converted them to globals, his attempt to substitute for cos(phi) wouldn't have worked.

@rlopez Note that option output= integral is available for Student:-VectorCalculus:-Flux, but, oddly, not for :-VectorCalculus:-Flux. The equivalent option for the latter command is simply inert.

@Al86 You need to remove the restart from the middle of your worksheet. Then vv's Answer works.

@Al86 Your uploaded code still won't download. My guess is that MaplePrimes can't handle file names with plus signs. Please try again with a different name.

@necron The first operand of an unevaluated integral is the integrand. For example, if

J:= Int(x^2, x= 0..1);

then op(0, J) is Int, op(1, J) is x^2, and op(2, J) is x= 0..1.

In addition to what Tom said, capitalization is important: It's ShowSolution, not showsolution.

@shadi alomari 

Yes, it does appear that your package has no Maple-accessible help file. If it did, there'd be a file named *.hdb (for your Maple 13) or *.help (which'd only work in more-recent Maple). If there was a help file, you'd access it just like regular Maple help. 

@Preben Alsholm In addition to what was mentioned by Christopher, a reason to not prepend to libname is that it makes it impossible to override a pre-installed command with another of the same name. My warning---with its capital letters and exclamation points---makes it seem that prepending would be as disastrous as overwriting. I didn't mean that, so I updated my Reply to mollify the strength of my warning for the prepend case.

@acer Perhaps my comment was misconstrued. I didn't mean to imply that map[evalhf] was better than the prepackaged commands. I simply meant that map[evalhf] was better than plain map. I updated the Reply to make this more clear.