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GATE 2018 ME (ME1) – Question 37

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 2 marks · Multiple choice

For position vector $\mathbf r=x\hat i+y\hat j+z\hat k$ and outward normal n, evaluate $\oiint_S\mathbf r\cdot\mathbf n\,dS$ on the closed boundary S of volume V.

  1. V
  2. 2V
  3. 3V
  4. 4V

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Show answer and explanation

Correct answer: (C) 3V

Explanation

Position-vector divergence is $\nabla\cdot\mathbf r=1+1+1=3$. The divergence theorem gives $\oiint_S\mathbf r\cdot\mathbf n\,dS=\iiint_V3\,dV=3V$ (C). The visually printed final component is z; native extraction incorrectly repeats y.