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  1. GATE2019-2-102
  2. GATE2019-2-102
  3. GATE2019-2-102
  4. GATE2019-2-102

Given a vector $\overrightarrow{u} = \frac{1}{3} \big(-y^3 \hat{i} + x^3  \hat{j} + z^3 \hat{k} \big)$ and $\hat{n}$ as the unit normal vector to the surface of the hemipshere $(x^2+y^2+z^2=1; \: z \geq 0)$, the value of integral $ \int (\nabla \times \overrightarrow{u}) \bullet \hat{n} \: dS$ evaluated on the curved surface of the hemishepre $S$ is

  1. $- \frac{\pi}{2}$
  2. $\frac{\pi}{3}$
  3. $\frac{\pi}{2}$
  4. $\pi$
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