Answer
See proof
Work Step by Step
Step 1: In this problem, we have to explain the following: why is it necessary for a conservative vector field, to be normal to a circle, on at least two points.
Step 2: In this problem, the following information is given: (1) \( \mathbf{F} \) is a two-dimensional vector field (2) that has zero divergence at all points (3) that has zero curl at all points Note: (2) and (3) are due to the conservative nature of the field
Step 3: In order to identify the divergence at a point, we draw a small circle around it, and treat the vector fields as the flow of water: (1) if the net flow of the water is positive inside that circle, then the divergence is positive (2) if the net flow of the water is negative inside that circle, then the divergence is negative In this case, the divergence and curl are zero, everywhere. This means that all the vector fields point in the same direction Since \( \mathbf{C} \) is a circle, and \( \mathbf{F} \) points in the same direction at all points, there are two points such that: the vector fields have the same direction, as the diameter through those points
