@sand15 

...how the shortest paths between two points on closed surfaces in R^3 can be calculated and visualized using Maple. This refers to closed surfaces with a "gender" > 0. Simple examples of such surfaces are the torus and the pretzel (a figure eight made of tubes). Unfortunately, I only know some theory about it :-( .

@nm 

...that had slipped my mind. However, I did ask for a "closed" solution there.

@vv 

...that ("Order") really helped :-) .

@dharr 

...I was looking for exactly the kind of command used in "sol3".

@nm 

... possible in (3) or (4)?

@vv 

... ...suggests the following regarding how Maple operates:
In principle, when calculating the indefinite integral, the integrand is treated as a complex-valued (possibly holomorphic) function - even in the case of real-valued functions, where the imaginary part is defined as identically zero. The discontinuity of the real part in this instance is, in my view, highly interesting. Many thanks for this insight :-) .

@janhardo 

...It is impressive how Maple implements this. A loud "knock-knock" on the seminar table :-)

@janhardo 

...on the result?

@acer 

...of the difference curve is, logically speaking, a solution. This makes the speed of convergence numerically apparent when the sequence of differences between successive terms is examined numerically (along the lines of Cauchy, for instance). And after a lot of theory, I am currently primarily interested in the numerical behavior of convergent "objects."

So, there is no "magnifying glass" for this?

BTW
Your algorithm contains a number of commands that are new to me. I still have a lot to discover there :-) .

@dharr @ janhardo @nm

...for the many new possibilities. Exploring the world of Maple's "differential and integral calculus" is interesting and educational for me. And now I can continue to practice bridging the gap between theory and practical application. :-)

By the way:

Is there a command for integration by parts? That would be helpful with unwieldy expressions.

@Kitonum 

I can understand the logic behind this command structure. Learned something new again :-) .

@Christian Wolinski 

...I (a beginner in Maple) didn’t understand this. Just like a long time ago, I had thought of a so-called ‘do-loop’, as I remember it from the days of Algol and Fortran. This was supposed to generate and display a list of results for a selected index range. I would appreciate further help.

@Aliocha 

......a simple recurrence relation should be possible. Recalling an old exercise and deciphering my handwriting, it might look like this:   I(n+1)=2*I(n)-I(n-1)
As far as I can tell at a quick glance, this can be derived from @Rouben Rostamian's result. However, programming it in Maple is still too difficult for me as a beginner.

edited:

I overlooked a term in the recurrence relation. Using @Rouben Rostamian's notation, the recurrence is:
f(n+1,x)=2*f(n,x)-f(n-1,x)+(2/n)*sin(n*x)

@Mariusz Iwaniuk 

...because countably many solutions to the equation become visible.

@janhardo 

...given Green's theorem, the structure of the integrand in the problem's surface integral looked so tempting to me - appearing as the difference of two rational terms where the degree of the denominator is two higher than that of the numerator :-(
Your analytical approach is impressive :-) . I've learned something new about Maple.