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GATE Mechanical 2022 Set 1 | Question: 3

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Given a function $\varphi =\frac{1}{2}\left (x ^{2} + y^{2} + z^{2} \right )$  in three-dimensional Cartesian space, the value of the surface integral $$\oint_{s}\hat{n}\cdot \triangledown \varphi dS,$$ where $S$ is the surface of a sphere of unit radius and $\hat{n}$ is the outward unit normal vector on $S$, is

  1. $4 \pi$
  2. $3 \pi$
  3. $\dfrac{4 \pi}{3}$
  4. $0$
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