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GATE 2019 ME (ME2) – Question 36

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

For $\vec u=(-y^3\hat i+x^3\hat j+z^3\hat k)/3$, find $\int_S(\nabla\times\vec u)\cdot\hat n\,dS$ over the curved upper unit hemisphere $x^2+y^2+z^2=1$, $z\ge0$, with outward unit normal.

  1. $-\pi/2$
  2. $\pi/3$
  3. $\pi/2$
  4. $\pi$

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Correct answer: (C) $\pi/2$

Explanation

Curl is $(0,0,x^2+y^2)$. Stokes’ theorem lets us use the unit disk with upward normal. In polar coordinates the flux is $\int_0^{2\pi}\int_0^1r^3\,dr\,d\theta=\pi/2$. **C**. The sign matches the outward upper-hemisphere orientation.