GATE Mechanical 2022 Set 1 | Question: 3

Question

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

• A. $4 \pi$
• B. $3 \pi$
• C. $\dfrac{4 \pi}{3}$
• D. $0$