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GATE 2025 CE (CE2) – Question 21

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 1 mark · Multiple select

Consider a velocity vector, $\vec{V}$ in (x, y, z) coordinates given below. Pick one or more CORRECT statements(s) from the choices given below.
$\vec{V} = u\hat{x} + v\hat{y}$

  1. z-component of Curl of velocity; $\nabla \times \vec{V} = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\hat{z}$
  2. z-component of Curl of velocity; $\nabla \times \vec{V} = \left(\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}\right)\hat{z}$
  3. Divergence of velocity; $\nabla \cdot \vec{V} = \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)$
  4. Divergence of velocity; $\nabla \cdot \vec{V} = \left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)$

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Correct answer: (A) z-component of Curl of velocity; $\nabla \times \vec{V} = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\hat{z}$; (C) Divergence of velocity; $\nabla \cdot \vec{V} = \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)$

Explanation

For a two-dimensional field the only component of the curl is $(\nabla \times \vec{V})_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$ (A is correct, B has the wrong form). The divergence is $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$ (C is correct, D mixes up the derivatives).