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GATE 2024 CE (CE1) – Question 37

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 2 marks · Multiple choice

A vector field $\vec{p}$ and a scalar field $r$ are given by
$\vec{p} = (2x^2 - 3xy + z^2)\hat{i} + (2y^2 - 3yz + x^2)\hat{j} + (2z^2 - 3xz + x^2)\hat{k}$
$r = 6x^2 + 4y^2 - z^2 - 9xyz - 2xy + 3xz - yz$
Consider the statements P and Q.
P: Curl of the gradient of the scalar field $r$ is a null vector.
Q: Divergence of curl of the vector field $\vec{p}$ is zero.

Which one of the following options is CORRECT?

  1. Both P and Q are FALSE
  2. P is TRUE and Q is FALSE
  3. P is FALSE and Q is TRUE
  4. Both P and Q are TRUE

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Correct answer: (D) Both P and Q are TRUE

Explanation

For any scalar field with continuous second derivatives, $\nabla \times (\nabla r) = \mathbf{0}$, so P is true. For any vector field with continuous second derivatives, $\nabla \cdot (\nabla \times \vec{p}) = 0$, so Q is true. Both fields here are polynomials, so these identities hold.