1. The problem is to understand the meaning and calculation of the operator \nabla \times, known as the curl of a vector field. 2. The curl of a vector field $ \mathbf{F} = (F_x, F_y, F_z) $ in three-dimensional space is defined as: $$\nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)$$ 3. This operator measures the rotation or the swirling strength of the vector field at a point. 4. To compute it, take the partial derivatives of the components of $ \mathbf{F} $ as shown in the formula. 5. The curl is a vector that points in the direction of the axis of rotation, and its magnitude corresponds to the strength of rotation. 6. Important rules: - The curl of a gradient field is always zero. - The curl operator is only defined in three dimensions. 7. Example: If $ \mathbf{F} = (y, -x, 0) $, then $$\nabla \times \mathbf{F} = \left( \frac{\partial 0}{\partial y} - \frac{\partial (-x)}{\partial z}, \frac{\partial y}{\partial z} - \frac{\partial 0}{\partial x}, \frac{\partial (-x)}{\partial x} - \frac{\partial y}{\partial y} \right) = (0 - 0, 0 - 0, -1 - 1) = (0, 0, -2)$$ This shows the curl vector points in the negative z-direction with magnitude 2.