For vector field V=a i+b j+c k, identify the invalid differential operation.

$\nabla\cdot(\nabla\times\mathbf V)$

$\nabla\times\nabla(\mathbf V)$

GATE 2022 BM – Question 13

Engineering Mathematics · Calculus · 1 mark · Multiple choice

For vector field $\mathbf V=a\hat i+b\hat j+c\hat k$, identify the invalid differential operation.

$\nabla\cdot(\nabla\times\mathbf V)$
$\nabla\times(\nabla\cdot\mathbf V)$
$\nabla(\nabla\cdot\mathbf V)$
$\nabla\times\nabla(\mathbf V)$

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Correct answer: (B) $\nabla\times(\nabla\cdot\mathbf V)$

Explanation

Divergence of a vector is a scalar. Taking its curl, as in B, is not a defined scalar-to-vector operation in ordinary three-dimensional vector calculus. Divergence of curl and gradient of divergence are defined. The source’s D notation can be interpreted componentwise as curl of each component gradient; the official invalid choice is B.

GATE 2022 BM – Question 13

Explanation

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