GATE 2015 EC – Question 37

Engineering Mathematics · Vector Analysis · 2 marks · Multiple choice

A vector $\vec{P}$ is given by $\vec{P} = x^3 y \vec{a}_x - x^2 y^2 \vec{a}_y - x^2 yz \vec{a}_z$. Which one of the following statements is TRUE?

$\vec{P}$ is solenoidal, but not irrotational
$\vec{P}$ is irrotational, but not solenoidal
$\vec{P}$ is neither solenoidal nor irrotational
$\vec{P}$ is both solenoidal and irrotational

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Correct answer: (A) $\vec{P}$ is solenoidal, but not irrotational

Explanation

The divergence is $3x^2 y - 2x^2 y - x^2 y = 0$, so the field is solenoidal. The $z$-component of the curl is $\frac{\partial P_y}{\partial x} - \frac{\partial P_x}{\partial y} = -2xy^2 - x^3$, which is not zero, so it is not irrotational.