Determine if the vector fields shown in Figure 1 and 2 are c

Determine if the vector fields shown in Figure 1 and 2 are conservative or not. Justify your answer.

Solution

Solution: It is usually easy to determine that a given vector field is not conservative. If there exist a closed path around which the circulation of the vector field doesn\'t vanish.

Rule to determine whether a given vector field is conservative.

A Vector field will be conservative if it satisfies the following rules:

(1) The level curves must be everywhere perpendicular to the vector field.

(2) The level curves must be close together where the magnitude of the vector is large.

(3) Level curves corresponding to different values may not intersect.

Figure 1: If we draw the level curves we see that level curves are not perpendicular to the vector field. Hence the vector field in figure 1 is not conservative.

Figure 2: If we draw the level curves we see that level curves are not perpendicular to the vector field. Hence the vector field in figure 2 is also not conservative.