Generating an Array of Vectors

Question

I'm working a lot with nested lists of $(x,y)$ coordinates (of a FEM-like mesh, but that's not important), like so:

nodes = { {{0,0}, {0,1}, {0,2}}, {{1,0}, {1,1}, {1,2}} }

This has the nice convenient structure that nodes[[i,j]] gives me the $(x,y)$ coordinates of my "$(i,j)$th" node, so that nodes[[i,j]][[1]] is my $x$ coordinate and nodes[[i,j]][[2]] is my $y$ coordinate. This works nicely for my purposes.

How would I construct a variable size array of this kind using array[] et. al? If I use for instance

nodes = array[array[X,2],{4,4}]

I get a 4-by-4 array where nodes[[i,j]] is of the form {X[1],X[2]}[1,1]. This is close to what I want, and I could change a bunch of my code to work with this, but I'd rather not - I'd love to have something where nodes[[i,j]] looks like this:

{X[i,j][1], X[i,j][2]}

The reason I'd like to do this is I have defined some functions that take an array of nodes as input and computes some quantity, i.e. F[nodes_]:= (*something*), and I'd like to compute derivatives of this function with respect to each of its variables, i.e. $F_{x_i}$ or $F_{y_j}$.

I've tried various things with no success, and I've also tried looking for a similar construct in the documentation and here on SE, but I didn't see anything. Any ideas?

Edit: I had too many square braces in some of my code before. Fixed.

Answer 1

A simpler way to give an array whose $(i,j)$ element is {X[i,j][1], X[i,j][2]} is to just use Table:

m = 3;
n = 4;
Table[X[i, j][k], {i, m}, {j, n}, {k, 2}]

Output is:

{{{X[1, 1][1], X[1, 1][2]}, {X[1, 2][1], X[1, 2][2]}, {X[1, 3][1], 
   X[1, 3][2]}, {X[1, 4][1], X[1, 4][2]}}, {{X[2, 1][1], 
   X[2, 1][2]}, {X[2, 2][1], X[2, 2][2]}, {X[2, 3][1], 
   X[2, 3][2]}, {X[2, 4][1], X[2, 4][2]}}, {{X[3, 1][1], 
   X[3, 1][2]}, {X[3, 2][1], X[3, 2][2]}, {X[3, 3][1], 
   X[3, 3][2]}, {X[3, 4][1], X[3, 4][2]}}}

Is this what you are looking for?

Answer 2

I believe this does what you requested:

nodes = Array[X[#, #2][#3] &, {7, 7, 2}];

Check:

nodes[[2, 3]]
nodes[[5, 4]]
nodes[[7, 1]]

{X[2, 3][1], X[2, 3][2]}
{X[5, 4][1], X[5, 4][2]}
{X[7, 1][1], X[7, 1][2]}