91Ó°ÊÓ

The divergence of a vector field F(x, y) = (P(x, y), Q(x, y)) is the scalar function that measures the tendency of the field to "spread out" or "converge" at a point. The mathematical formulation of divergence is given as: $$ \text{div } F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} $$ Our task is to find a vector field F(x, y) that has zero divergence everywhere in the plane, meaning the above equation should equal zero for all points (x, y) in the plane.

One simple example of a vector field with zero divergence everywhere in the plane is F(x, y) = (y, -x). Here, P(x, y) = y and Q(x, y) = -x. Let's check that the divergence of this field is zero everywhere: $$ \text{div } F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = \frac{\partial y}{\partial x} + \frac{\partial (-x)}{\partial y} = 0 + 0 = 0 $$ So, F(x, y) = (y, -x) is a suitable example of a vector field with zero divergence everywhere in the plane.

To sketch the vector field F(x, y) = (y, -x), we can place a vector with components (y, -x) at each point (x, y) in the plane. We can notice a pattern in the vector field: the vectors are pointing in a counterclockwise circular direction. This makes sense because the rotation of a vector field F(x, y) = (y, -x) equals the negative derivative of P(x, y) with respect to x (i.e., -y) minus the negative derivative of Q(x, y) with respect to y (i.e., -x) at each point (x, y). The circular pattern ensures every time that the divergence of the field is zero everywhere in the plane. So, the sketch of the vector field F(x, y) = (y, -x) should depict vectors pointing in a counterclockwise circular direction around the origin where the length of the vector increases as we move away from the origin.