Note that the "kernel connection" message, while correct in a very narrow sense, is also very misleading, because the vast majority of times that you get that message are due to the kernel crashing due to its own internal bugs. If you get this message during an active Maple session (as opposed to a session that has just started), I'd guess that the chance that the problem has anything to do with firewalls is less than 1 in 10,000.

@Joe Riel Is your proposed orseq any better than ormap?

@BOverdorp Erasing all variables is an unusual thing to do, although if you insist, I could give you a command for it. Almost certainly you should try a different approach. The .mpl file should contain a procedure with parameter H. You read that file once, before entering the loop. The statement in the loop should be an invocation of the procedure. The local variables of the procedure are automatically re-initialized each time the procedure is run.

@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.