Question

Consider the velocity field \[ \vec{V} = (2x + 3y)\hat{i} + (3x + 2y)\hat{j}. \] The field \( \vec{V} \) is:

• A. divergence-free and curl-free
• B. curl-free but not divergence-free
• C. divergence-free but not curl-free
• D. neither divergence-free nor curl-free

Hint:

A vector field with zero curl is irrotational; one with zero divergence is incompressible.

Answer 1

The Correct Option is B

Step 1: Compute divergence.
\[ \nabla \cdot \vec{V} = \frac{\partial}{\partial x}(2x + 3y) + \frac{\partial}{\partial y}(3x + 2y) = 2 + 2 = 4. \] Since divergence is \(4 \neq 0\), the field is **not divergence-free**.

Step 2: Compute curl (2D).
\[ (\nabla \times \vec{V})_{z} = \frac{\partial}{\partial x}(3x + 2y) - \frac{\partial}{\partial y}(2x + 3y) = 3 - 3 = 0. \] Curl is zero → field is **curl-free**.

Step 3: Conclusion.
Thus, the field is **curl-free but not divergence-free**.