### Vector Fields and Line integrals

#### Part 3: Experiments with Work and Vector Fields

In this portion of the module, you will apply what you have learned about work and vector fields to explain the results of some experiments.

The applet below allows you to select a vector field (by clicking on one of the numbers) and to draw a curve. As you draw, the curve appears in green, and arrows representing the outputs of the vector field at points on the curve appear in yellow. If you draw more slowly and carefully, there will usually be more yellow arrows drawn on the green curve, which may give you a clearer picture of what is going on.

The purple number that appears in the upper left hand corner of the applet gives the amount of work that has been done so far, if the force is represented by the vector field and the path that the object is moved along is the green curve you have traced out.

1. Under what circumstances is it possible for an object to move and no work to be done?
2. For each of the seven different vector fields, decide whether there are any curves along which an object could move without any work being done. If you believe that any of the vector fields have such curves, describe the curves in words, and then use the applet to test your predictions.
3. What is the geometrical relationship between the curve(s) you have found and the vectors in the vector fields?
4. Next, select a particular vector field from the seven you have available. You can use any of the vector fields that you like, but it important to choose one and stick with it for this question. Using the applet either as a tool for exploration or as a way of checking your answers, describe (in words) closed* curves that give

• a large, positive amount of work;
• a small, positive amount of work;
• a small, negative amount of work;
• a large, negative amount of work.

*Note: A closed curve is one that starts and ends at the same point.

5. For two of the curves that you have found, explain why the shape of the curve and the arrangement of vectors in the vector field combine to give the amount of work (large positive, small positive, small negative, or large negative) that you predict.
6. Next, choose one of the other vector fields in the applet. Working from the appearance of the vector field alone, describe (in words) the appearance of closed curves that give

• a large, positive amount of work;
• a small, positive amount of work;
• a small, negative amount of work;
• a large, negative amount of work.

Use the applet to check your predictions.