Topic: Engineering Mathematics
The vector field $\vec{F}=x\hat{i}-y\hat{j}$ (where i and $\hat{j}$ unit vectors) is-
- A. Divergence free, but not irrotational
- B. irrotational, but not divergence free
- C. Divergence free and irrotational
- D. Neither divergence free nor irrotational
Answer: C. Divergence free and irrotational
Explanation: Divergence of $\vec{F} = \nabla \cdot \vec{F} = \frac{\partial x}{\partial x} + \frac{\partial (-y)}{\partial y} = 1 - 1 = 0$ (so it is divergence free). Curl of $\vec{F} = \nabla \times \vec{F} = \hat{k}(\frac{\partial (-y)}{\partial x} - \frac{\partial x}{\partial y}) = \hat{k}(0 - 0) = 0$ (so it is irrotational).