@tomleslie Command-line Maple won't run worksheets, but it will run .mpl files. The .mpl file to be run can be put on the command line that invokes cmaple in the manner that the OP was trying (see ?maple). Like I told the OP, this is not the solution to his problem, but it is occasionally useful. Indeed, I use something like this to update archives (libraries).

@one man The new surface plots look excellent, and I think that you can how see much superior they are compared to those produced with implicitplot3d. Did you use plot3d? You don't show any Maple code, so I don't know exactly what you mean by "array" and "matrix". If you haven't done so already, I should be able to take your procedure and turn it into something that can be used with plot3d in the form

plot3d([X, Y, Z], a..b, c..d);

@one man A parametrization of a surface is a function from R^2 to R^3 (with some smoothness requirements). We can safely ignore the smoothness requirements, but we need the function. If you could provide a purely numeric procedure of two real arguments that returns a list of three reals, that would be a great achievement and a facilitator of 3D plotting. Perhaps you could take what you have done with curves and do some interpolating. See ?CurveFitting,ArrayInterpolation. 

Would you please show some code, preferably with some plots, that shows the behaviour that you're talking about?

@mskalsi Yes, you want a power series in xi; that makes sense. Here's an easier way:

series(%, xi);

2*P[1]*P[0]+P[2]*P[3]+P[1]+(4*P[0]*P[2]+2*P[1]^2+2*P[4]*P[5]+2*P[2])*xi+(6*P[0]*P[3]+6*P[1]*P[2]+3*P[6]*P[7]+3*P[3])*xi^2+(8*P[0]*P[4]+8*P[1]*P[3]+4*P[2]^2+4*P[8]*P[9]+4*P[4])*xi^3+(10*P[0]*P[5]+10*P[1]*P[4]+10*P[2]*P[3]+5*P[10]*P[11]+5*P[5])*xi^4+(12*P[1]*P[5]+12*P[2]*P[4]+6*P[3]^2)*xi^5+O(xi^6)

coeff(convert(%, polynom), xi, 2);

@mskalsi Sorry, I forgot that the denominators contain xi also. So the command for the coefficient of xi^2 is

coeff(eval(%, infinity= 3), xi, 2);

which returns what you have above.

Asking for the "n=2" term is still nonsense, but I gave the command for it anyway. You didn't comment on that.

@Markiyan Hirnyk The example that you present is analytic in all coordinates, so it doesn't present the problem that Mac Dude is referring to.

All the definite integrals will be done if option continuous is used:

int(exp(x^3), x= 1..2, continuous);

I've seen this bug before, for other integrands. The antiderivative that you got seems correct (I've checked it numerically and symbolically), so it can be applied to the definite integrals. The following returns unevaluated also (with or without) FTOC:

int(exp(x^3), x= 0..X) assuming X > 0;

There's a miniscule anomaly with the antiderivative: To be evaluated at 0, you need to take the limit, which is 0.

@fadhli If you can supply an algorithm, I can probably translate it into Maple. Note that I have very little knowledge of the underlying mathematics. I'm just very good at translating algorithms into Maple.

@jcrook There is a problem with your statement of the problem, in the range of theta: 0 <= theta <= Pi makes a complete loop of the curve; 0 <= theta <= 2*Pi goes around twice. It might make sense to ask what is the arc length in two revolutions, but it doesn't make sense to me to ask about the area of two revolutions.

Also, in your arc length integral formula, the derivative needs to be squared.

@raazia The roots that you seek are BesselJZeros(0,n)/R.

@Giulianot There are memory limitations also. I just killed the dsolve because it was using 9 Gig of memory, which is all that could devote to it.

Certainly the average computer these days can tolerate an uptime of a few days. Yes, you need to set it so that it doesn't turn off after a certain amount of idle time.

But I don't have much hope for your system. Like I said, dependent variables that appear in the denominators of exponents are a nasty thing.

Use Google's site operator to do a site-specific search. For example

"Maple 6..9" site:www.maplesoft.com/applications

brings up 566 hits.

@Giulianot Thanks, the system is much easier to work with now.

My suspicion is that it's just too complicated. How long have you let it run? I would let it run a few days before giving up. There are four methods available for DAEs: rkf45_dae, ck45_dae, rosenbrock_dae, or mebdfi. For example, I gave the commands:

sys:= {...your long system...}:
dsolve(sys, numeric, method= rosenbrock_dae);

and it's currently running on my machine, although I don't have a few days to devote to it.