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GATE 2021 AG – Question 11

Engineering Mathematics · Vector Calculus · 1 mark · Multiple choice

For differentiable $\mathbf v=v_1\hat i+v_2\hat j+v_3\hat k$, choose the Cartesian expression for curl v.

  1. $(\partial_zv_2-\partial_yv_3)\hat i+(\partial_xv_3-\partial_zv_1)\hat j+(\partial_yv_1-\partial_xv_2)\hat k$
  2. $(\partial_yv_3-\partial_zv_2)\hat i+(\partial_xv_1-\partial_zv_3)\hat j+(\partial_yv_2-\partial_xv_1)\hat k$
  3. $(\partial_yv_3-\partial_zv_2)\hat i+(\partial_zv_1-\partial_xv_3)\hat j+(\partial_xv_2-\partial_yv_1)\hat k$
  4. $(\partial_yv_2-\partial_zv_3)\hat i+(\partial_xv_3-\partial_zv_1)\hat j+(\partial_xv_1-\partial_yv_2)\hat k$

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Correct answer: (C) $(\partial_yv_3-\partial_zv_2)\hat i+(\partial_zv_1-\partial_xv_3)\hat j+(\partial_xv_2-\partial_yv_1)\hat k$

Explanation

Expand the formal determinant with first row unit vectors, second row ∂x,∂y,∂z, and third row v₁,v₂,v₃. Cofactor signs give $(\partial_yv_3-\partial_zv_2,\partial_zv_1-\partial_xv_3,\partial_xv_2-\partial_yv_1)$. Thus **C**.

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