The vector \vecA=(2x+1)\hati+(x^2-6y)\hatj+(xy^2+3z)\hatk is a

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Question Id : 11662 | Context :NIMCET 2024

Question

The vector

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

🎥 Video solution / Text Solution of this question is given below:

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.

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Question

Vector Field:

Divergence:

Curl:

Final Answer:

Vector Sink Field Analysis

Divergence:

✅ Conclusion:

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