Thanks, yes this definitely became an overhaul of the procedure ( I didn't expect )
I definitely need to study this : how some things have been changed and why.
What is the objection to using a function operator instead of an expression notation ?
This procedure is still for simple surfaces it appears
For more complicated surfaces it becomes f(x,y,z) and the domain becomes a space curve
I still want to try to extend the existing procedure... 

If this should work the VisualizeFunctionAndCurve () procedure, then a further modification for the domain curve  

A plane curve can be represented in two ways: by an equation H(x,y)=0 (the implicit description) and parametrically by a pair of equations with one parameter (the parameter description).

@sija 
Thanks, i will look at it again

Interesting, but from the looks of it, none of the animations work in Maple 2024, or am I mistaken? 

Have revisited the task and have the idea that this elaboration could be better in Maple ? 

Download lijnintegraal_2e_poging_maple_primes.mw

@Carl Love 
I asked the AI for an alternative to the is () command and it didn't come up with a thoughtful answer anyway.
Fine to is ( ) command and apparently there are no alternatives to this?
I find it tricky with that evaluation in Maple with the cauchy-riemann equations.

@janhardo 
Can I also do that check on the cauchy riemann equations via : evalb ?

if evalb(u_x = v_y) and evalb(u_y = -v_x) then
    printf("The function is analytic (holomorphic) at this point.\n");
    printf("The derivative f'(z) is %a + I*%a\n", u_x, v_y);
else
    printf("The function does not satisfy the Cauchy-Riemann equations and is not analytic.\n");
end if

@Carl Love 
Thanks, Yes, that "is" command is essential to establish that the cauchy-riemannt equations are true for all variables

@Axel Vogt 
Thanks, forms a nice basis to rewrite the procedure and the hereby use the D operator instead of diff command.

@acer 
Yes, when to use a branch from a topic?
There are several ways to derive that zeta(2) value. 
I thought this was 1 branch to explore this further and another branch for a different approach to derive zeta( 2).
Otherwise it will be one long thread of proof methods topics , fine by me too and forget the branches.

How about checking this with ODESteps in Student package ?

@vv 

Thanks, I think you are absolutely right to study the methods of evidence.
The difficulty for me is still how to start translating this into Maple, and your elaboration is an example of how it can be done.

Don't make any progress yet , let me ask Maple AI to come up with the code variants to solve the Basel problem ( The Zeta (2) value ). 
Note: get the impression that the standard AI 3.5 works better for Mathematica ? (not the plugin yet ) 
May try this for the Basel problem for Maple as well...

@janhardo 

It is about comparing series expansion and product expansion of Sin[x]/x. The coefficient of x^2 in the series must coincide with coefficient of x^2 in the product (after expanding the product and collecting powers of x).
Seems to me one of the proofmethods...

simply this equation?
 

eq1 := Product(1 - x^2/(n^2*Pi^2), n = 1 .. infinity) = Sum((-1)^n*x^(2*n)/(2*n+1)!, n = 0 .. infinity) = sin(x)/x;

@sand15 
 

"More generally, ChatGPT does not shine for its clarity (Point 6 is quite obscure)."

You can challenge chatGPT to come up with a better answer and also zoom in on a detail 
I'm using the public chatGPT 3.5 (free) and not the one in Maple itself and therein is a better version chatGPT 4.5?, but don't find it working conveniently in Maple yet 
Also mathematica has developed its own language model plugin and that requires chatgtp 4.5 with a subscription
I wonder how this works in Mathematica 
Ideal would be if Maple would use its own language model focused on Maple content and then you can ask anything, at least it seems to me that this is possible ?