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GATE 2022 ME (ME1) – Question 13

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 1 mark · Multiple choice

Given a function $\varphi = \frac{1}{2}(x^2 + y^2 + z^2)$ in three-dimensional Cartesian space, the value of the surface integral $\oint_S\hat{n} \cdot \nabla\varphi\,dS$, 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. $4\pi/3$
  4. 0

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Correct answer: (A) $4\pi$

Explanation

By the divergence theorem the integral equals $\iiint\nabla^2\varphi\,dV$. Here $\nabla^2\varphi = 3$, and the volume of the unit sphere is $\frac{4\pi}{3}$. The value is $3 \times \frac{4\pi}{3} = 4\pi$.