Programming with Dynamic

I don't understand the following:

  f[a_, b_] := a + b
  ls = {1, 2, 3};
  MapThread[f, {ls, {10, 20, 30}}]

This yields ( as expected ) {11,22,33}

If I change the code to

  f[a_, b_] := a + b
  ls = {1, 2, 3};
  MapThread[f, {Dynamic[ls], {10, 20, 30}}]

mma returns the following message:

  MapThread::mptd: Object {1,2,3} at position {2, 1} in MapThread[f,{{1,2,3},{10,20,30}}] has only 0 of required 1 dimensions. >>

Question: Why does this happen? How can it be fixed?

Answer

Even though Dynamic[ls] is presented as ls, its head is Dynamic:

x = Dynamic[ls]
Head[x]
(*
{1, 2, 3}
Dynamic
*)

or

Basically, Dynamic is a wrapper, and is there even if you cannot see it. Its like you were writing

MapThread[f, {symb, {10, 20, 30}}]

with symb undefined, or

MapThread[f, {Integrate[symb[var], var], {10, 20, 30}}] 

You can fix it by doing

Dynamic[MapThread[f, {ls, {10, 20, 30}}]]

instead.

So:

Wherever you had written Dynamic[ls], you still have Dynamic[ls]. It may be displayed by the frontend as {1,2,3}, but it is not; it's Dynamic[ls]. So, MapThread goes to look at the first part of its second argument and sees Dynamic instead of a List, so it stops and emits a message; this message includes Dynamic[ls] in it. When the message reaches the "surface" to be displayed, Dynamic[ls] gets automatically interpreted by the frontend as usual: it's displayed as {1,2,3}, because that's what ls evaluates to. But it is still Dynamic[ls], not {1,2,3}, which is simply how it is displayed.

Try this: Do[Sin[i], Dynamic@{i, 1, 2}] and think about what happens...