1. The problem asks to identify the condition for a vector \( \mathbf{V} \) to be irrotational. 2. By definition, a vector field \( \mathbf{V} \) is irrotational if its curl is zero. 3. The curl of a vector field \( \mathbf{V} = (V_x, V_y, V_z) \) is given by: $$\nabla \times \mathbf{V} = \left( \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z}, \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x}, \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y} \right)$$ 4. If \( \nabla \times \mathbf{V} = \mathbf{0} \), then \( \mathbf{V} \) is irrotational. 5. The divergence \( \nabla \cdot \mathbf{V} \) being zero is a different condition related to incompressibility, not irrotationality. 6. Therefore, the correct condition for \( \mathbf{V} \) to be irrotational is: $$\text{curl } \mathbf{V} = 0$$ Final answer: curl \( \mathbf{V} = 0 \).