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GATE 2026 PH – Question 14

Mathematical Physics · Linear vector spaces: basis, orthogonality and completeness · 1 mark · Multiple choice

For a scalar field $\psi(\mathbf r)$ and a vector field $\mathbf A(\mathbf r)$, $\nabla\times(\mathbf A\psi)$ is equivalent to the expression

  1. $\psi(\nabla\times\mathbf A)-\mathbf A\times\nabla\psi$
  2. $\psi(\nabla\times\mathbf A)+\mathbf A\times\nabla\psi$
  3. Null vector
  4. $\mathbf A\times\nabla\psi-\psi(\nabla\times\mathbf A)$

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Show answer and explanation

Correct answer: (A) $\psi(\nabla\times\mathbf A)-\mathbf A\times\nabla\psi$

Explanation

Use the product rule for the curl of a scalar times a vector:
$$\nabla\times(\psi\mathbf A)=\psi\,(\nabla\times\mathbf A)+(\nabla\psi)\times\mathbf A .$$

(In components: $[\nabla\times(\psi\mathbf A)]_i=\epsilon_{ijk}\partial_j(\psi A_k)=\psi\,\epsilon_{ijk}\partial_jA_k+\epsilon_{ijk}(\partial_j\psi)A_k$.)

The cross product is anticommutative: $(\nabla\psi)\times\mathbf A=-\mathbf A\times\nabla\psi$. Therefore
$$\nabla\times(\mathbf A\psi)=\psi(\nabla\times\mathbf A)-\mathbf A\times\nabla\psi\quad(\text{option A}).$$