Here is a very nice example from The Student's Introduction to MATHEMATICA.

Manipulate[Module[{x, y},
  ContourPlot[Exp[-x^2 - y^2] + x y, {x, -1, 1}, {y, -1, 1},
   Contours -> 20,
   Epilog -> 
    Dynamic[{Arrow[{pt, 
        pt + {y - 2 E^(-x^2 - y^2) x, x - 2 E^(-x^2 - y^2) y} /. {x ->
            pt[[1]], y -> pt[[2]]}}]}]
   ]],
 {{pt, {.5, .5}}, Locator, 
  Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]

Which produces a wonderful demonstration.

My next step was to try the following:

Clear[x, y, f];
f = E^(-x^2 - y^2) + x y;
Grad[f, {x, y}];

Then in the next cell, I tried:

Manipulate[Module[{x, y},
  ContourPlot[f, {x, -1, 1}, {y, -1, 1},
   Contours -> 20,
   Epilog -> 
    Dynamic[{Arrow[{pt, 
        pt + Grad[f, {x, y}] /. {x -> pt[[1]], y -> pt[[2]]}}]}]
   ]],
 {{pt, {.5, .5}}, Locator, 
  Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]

Which gave only this:

I gave Evaluate a try but that didn't work. So I tried removing the Module.

Manipulate[
 ContourPlot[f, {x, -1, 1}, {y, -1, 1},
  Contours -> 20,
  Epilog -> 
   Dynamic[{Arrow[{pt, 
       pt + Grad[f, {x, y}] /. {x -> pt[[1]], y -> pt[[2]]}}]}]
  ],
 {{pt, {.5, .5}}, Locator, 
  Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]

And that worked.

But try typing

x=12

in the next cell and watch what happens to the arrow.

Finally I tried wrapping everything with a DynamicModule to see if it would prevent the x=12 issue in the notebook. First, this cell.

Clear[x, y, f];
f = E^(-x^2 - y^2) + x y;
Grad[f, {x, y}];

Then:

DynamicModule[{x, y},
 Manipulate[
  ContourPlot[f, {x, -1, 1}, {y, -1, 1},
   Contours -> 20,
   Epilog -> 
    Dynamic[{Arrow[{pt, 
        pt + Grad[f, {x, y}] /. {x -> pt[[1]], y -> pt[[2]]}}]}]
   ],
  {{pt, {.5, .5}}, Locator, 
   Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]]

This only produced:

As folks know, there has been a lot of discussion on not using Module inside of Manipulate and this is probably a good example of why not, but there is a lot of stuff happening here that I don't understand and could use some discussion explaining some of the issues:

1. Why does Dynamic in the first code just update the arrow and not the contour plot.

2. Why doesn't f = E^(-x^2 - y^2) + x y; and Grad[f, {x, y}]; work in the second piece of code?

3. Why doesn't DynamicModule work in the last piece of code?

4. And what is the best way to protect the arrow if a student type x=12 in their notebook?