A direct translation into Maple of what you have written is

local D:= n->
   Matrix(
      ((n+1)$2),
      (i,k)->
         if i <> k then P[n](t[i])/P[n](t[k])/(t[i] - t[k])
         elif i = 1 then -n*(n+1)/2
         elif i = N+1 then n*(n+1)/2
         else 0
         end if
   )
:

I can't read your denominators on n*(n+1). I have guessed 2.

Your method of presentation shows disrespect for this forum and its members. If you can't come up with a better presentation, you should at least first ask for permission to post something so crude, and beg for the forgiveness of your readers.

When you make an assignment to F[1], it also changes the meaning of F, which becomes a table, not a vector field. If you want a new variable which displays as F with a subscript, use F__1 (that's two underscores). That is a completely new variable, with no connection to F. When any name is followed by [anything], the resulting structure has some relation to the parent name.

Regarding your title: Consider this simple example:

x:= 2:
x:= 1/(x-1);
x:= 1/(x-1);

Surely you can understand that it is correct that the second command works and that the third one returns an error although they are syntacticly the same command.

One way to do what you want (and it's certainly the easiest programmatic way) is make a Matrix whose first row is the headers and whose second row is the matrices. So, some of the elements of this Matrix are themselves matrices.

Display:= Matrix(2,3, [[`Matrix 1`, `Matrix 2`, `Matrix 3`]]):
Display[2,1]:= A1: Display[2,2]:= A2: Display[2,3]:= A3:
Display;

If A and B are vectors, matrices, arrays, etc., of the same size and shape, then A *~ B performs the elementwise multiplication.

How would you recommend that this feature be indexed in the help pages, since you seem to have tried to find it? It is listed under ?elementwise

In Maple, e^x is exp(x). There is no caret (^) in it.

See ?Student,Calculus1,VolumeOfRevolution and ?Student,Calculus1,VolumeOfRevolutionTutor

A surface integral is a method of computing the flux (or flow) of a vector field (such as a force field or a moving fluid) through a surface. It is not ordinarily used to compute volume.

If you parametrize a surface, then in any integral that uses that parametrization, you need to multiply the integrand by the determinant of the Jacobian (the matrix of partial derivatives) of the transformation. This is completely analogous to changing dx to du when doing a u-substitution in a single integral. This determinant is where the extra factor of r comes from for polar and cylindrical coordinates and the extra factor of rho^2*sin(theta) for spherical coordinates. You haven't done this in your integral. I'm not saying that you should go do it! Because the easiest way to do this volume integral is in cylindrical coordinates.

Your plot is pretty good. It definitely shows the correct two surfaces.

In the worksheet below, I compute the volume five different ways, all using integrals. The Answer by Tom Leslie is incorrect.
 

Download SphereCylinder.mw

You should state your Maple version in your Question header.

It works for me in Maple 2016:

restart:
gg:=A* exp( - ( (t - t0) / (tau) )**2 ):
val1:=int(gg, t=-x0..x1) assuming t0::real, tau::real, x0<x1;

val1 := (1/2)*erf((x0+t0)/tau)*A*sqrt(Pi)*tau-(1/2)*erf((-x1+t0)/tau)*A*sqrt(Pi)*tau

You should put your Maple version in your Question header.

It works for me in Maple 2016:

restart:
gg:=A* exp( - ( (t - t0) / (tau) )**2 ):
val1:=int(gg, t=-x0..x1) assuming t0::real, tau::real, x0<x1;

@John Fredsted If you get that message ("The resource that you're looking for..."), it almost certainly means that a Question has been changed to a Post (or vice versa) or the author has removed their own material and the Active Conversations index hasn't yet been updated (it takes about 15 minutes). Why it takes 15 minutes to update such a trivial index is beyond me.

People's fear of censorship on Internet fora is vastly exaggerated. I'm only aware of two cases of censorship here at MaplePrimes in my years here. One was for some explicitly racist content, and the other was for posting some copyrighted code without permission. In both cases, the offending message was replaced with a message explaining the reason for removal.

Moderators have the ability to change Posts to Questions (or vice versa) or to edit content in any other way. I often change Posts to Questions or re-attach a Reply to a new parent. Indeed, I'm tempted to change this thread to a Post.

I wrote a Maple solution for this using the simplest code I could, and using only simple formulas from high-school algebra and geometry---no piecewise, integrals, etc. Let me know if you can understand this better.
 

Download Trapezoids.mw

It's done like this:

addcoords(
   EEspherical,
   [rho, theta, phi],
   [rho*cos(phi)*sin(theta), rho*sin(phi)*sin(theta), rho*cos(theta)]
);

Or, you can simply replace the existing spherical coordinates by using spherical instead of EEspherical.

You can add this command to an initialization file.

Assuming that you don't want to explicitly set the vertex positions and that you'll accept any reasonanly spread-out planar representation of the graph, it can be done like this:
 

Download DrawnGraph.mw

PolyhedralSets:-Plot(
   PolyhedralSets:-PolyhedralSet(
      {-x-y+3*z >= 0, -x+2*y >= 0, 3*x-2*y-z >= 0}
   )
);

Good Question; Vote Up.  You should do it like this:

Package:= module()
option package;
export 
   Output:= proc(Input)
      Record(
         'o1'= f1(Input)
         'o2'= f2(Input)
         'o3'= f3(Input)
      )
   end proc
;
end module:         

A Record is just a special simple type of module that only has exports, and there's no interaction amongst them. So, you would still access this as Output(Input):-o1.