The vector $\vec{A}=(2x+1)\hat{i}+(x^2-6y)\hat{j}+(xy^2+3z)\hat{k}$ is a

Vector Field:

\[ \vec{A} = (2x + 1)\hat{i} + (x^2 - 6y)\hat{j} + (xy^2 + 3z)\hat{k} \]

Divergence:

\[ \nabla \cdot \vec{A} = 2 - 6 + 3 = -1 \neq 0 \]

Not solenoidal ❌

Curl:

\[ \nabla \times \vec{A} = (2xy)\hat{i} - (y^2)\hat{j} + (2x)\hat{k} \neq \vec{0} \]

Not conservative ❌

Final Answer:

\( \vec{A} \) is neither conservative nor solenoidal.

Vector Sink Field Analysis

Given vector field:

\[ \vec{A} = (2x + 1)\hat{i} + (x^2 - 6y)\hat{j} + (xy^2 + 3z)\hat{k} \]

Divergence:

\[ \nabla \cdot \vec{A} = 2 - 6 + 3 = -1 \]

✅ Conclusion:

The divergence is negative at every point, so \( \vec{A} \) is a sink field.