Answer
See proof
Work Step by Step
Step 1: In this problem, we have to check for the following: Is it necessary for a conservative vector field to be normal to a square boundary.
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. So, if we consider a vector field in which all the vectors point along the diagonal of the square, then there will be no point at which the vector field is normal to the square.
