Continuation of a Problem with Manipulate

Question

For the beginning of this discussion, see: A problem with Manipulate.

Now, the final use of the Initialization command doesn't protect the Global workspace. What I am discussing is not a single Manipulate object. Rather, imagine a notebook filled with static code and many Manipulate objects.

Now, here is an example. Suppose that the values of a and b before performing the Manipulate are assigned as follows:

a = 1; b = 5;

Then you run the Manipulate object.

Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}], 
    Text["N = " <> ToString[n] <> ",    R = " <> 
      ToString[rightSum[n]], {(a + b)/2, f[b]}]}]], {{n, 10}, 10, 50, 
  10, Appearance -> "Labeled"},
 Initialization :> (a = 0; b = 1; dx[n_] := (b - a)/n; f[x_] := x^2;
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}])]

Now, you are back in your workspace and you enter:

In[37]:= a

Out[37]= 0

In[38]:= b

Out[38]= 1

So, you can see that the Manipulate code affected objects in the workspace. So, question 1 is, what do you add to this code to prevent this from happening.

Here is a second example. Supposed that in the workspace you have defined:

dx = (b - a)/n

Then you try to run the Manipulate object.

Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}], 
    Text["N = " <> ToString[n] <> ",    R = " <> 
      ToString[rightSum[n]], {(a + b)/2, f[b]}]}]], {{n, 10}, 10, 50, 
  10, Appearance -> "Labeled"},
 Initialization :> (a = 0; b = 1; dx[n_] := (b - a)/n; f[x_] := x^2;
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}])]

Now it doesn't work, sending error messages such as:

Coordinate {(4/$CellContext`n)[10], (4/$CellContext`n)[10]^2} should be a pair of numbers, or a Scaled or Offset form. Not only that, but my Documentation notebook is also messed up.

Here is a third problem. Suppose I change the function and a and b in the initialization section to create a new Manipulate. That is, I first do this: set a=0, b=1, and f[x_]:=x^2.

Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}], 
    Text["N = " <> ToString[n] <> ",    R = " <> 
      ToString[rightSum[n]], {(a + b)/2, f[b]}]}]], {{n, 10}, 10, 50, 
  10, Appearance -> "Labeled"},
 Initialization :> (a = 0; b = 1; dx[n_] := (b - a)/n; f[x_] := x^2;
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}])]

Then set a=0, b=2, and f[x_]=x:

Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}], 
    Text["N = " <> ToString[n] <> ",    R = " <> 
      ToString[rightSum[n]], {(a + b)/2, f[b]}]}]], {{n, 10}, 10, 50, 
  10, Appearance -> "Labeled"},
 Initialization :> (a = 0; b = 2; dx[n_] := (b - a)/n; f[x_] := x;
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}])]

If you enter both in a notebook, then evaluate the notebook, you will see that both graphs and values of a and b are the ones defined in the second Manipulate chunk.

So I need to learn more. This is similar to what I posted at: Difficulty with Manipulate that has same variables as other code lines in a notebook, but I obviously need to learn more.

What should be added to the Manipulate code to prevent these things from happening?

Answer 1

I would localize the variables inside the DynamicModule created by the Manipulate by using the ControlType -> None (or simply None for short) specification. This use is discussed here:

• What does None mean in a control specification for Manipulate?
• ControlType -> None Vs. Module inside Manipulate (i.e. making everything local)

Code:

Manipulate[
 Show[
  Plot[f[x], {x, a, b}, PlotStyle -> Thick, AxesLabel -> {"x", "y"}],
  Graphics[{
    Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}],
    Text["N = " <> ToString[n] <> ",    R = " <> 
      ToString[rightSum[n]], {(a + b)/2, f[b]}]
    }]],
 {{n, 10}, 10, 50, 10, Appearance -> "Labeled"},
 {{a, 0}, None}, {{b, 1}, None},
 {{dx, dx}, None}, {{f, f}, None},
 {{rightSum, rightSum}, None},
 Initialization :> (
   a = 0; b = 1;
   dx[n_] := (b - a)/n;
   f[x_] := x^2;
   rightSum[n_] := Total@Table[f[a + i dx[n]] dx[n], {i, 1., n}])]

Note that a declaration of the form {var, None} is equivalent to {{var, 0}, None}, so that the variable var would be initialized to 0. To prevent this, I initialized dx, f, and rightSum to their own symbols. If they were initialized to 0, then the definitions, dx[n_] := ... etc., would generate errors. (These initializations happen before the Initialization option is evaluated.)