Answer
The vector field $\mathbf{F}(x, y) = y\mathbf{i} - x\mathbf{j}$ represents a rotational field centered at the origin. Each vector points perpendicular to the radius vector from the origin, creating circular motion around it.
Work Step by Step
Explanation
- The $x$-component of the vector is $F_x = y$ and the $y$-component is $F_y = -x$.
- At any point $(x, y)$, the vector is tangent to a circle of radius $\sqrt{x^2+y^2}$ centered at the origin.
- The direction of rotation is clockwise because the vector field points in the direction $(y, -x)$.
- Such a vector field is divergence-free since $\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = 0 + 0 = 0$.
