GATE 2020 EC – Question 12

Engineering Mathematics · Vector Analysis · 1 mark · Multiple choice

For a vector field $\vec A$, which one of the following is FALSE?

$\vec A$ is solenoidal if $\nabla\cdot\vec A=0$.
$\nabla\times\vec A$ is another vector field.
$\vec A$ is irrotational if $\nabla^2\vec A=0$.
$\nabla\times(\nabla\times\vec A)=\nabla(\nabla\cdot\vec A)-\nabla^2\vec A$

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Correct answer: (C) $\vec A$ is irrotational if $\nabla^2\vec A=0$.

Explanation

A field is irrotational when $\nabla\times\vec A=0$, not when its Laplacian vanishes, so C is false. A, B and D are standard: the definition of solenoidal, the curl being a vector field, and the curl-of-curl identity.