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GATE 2023 MT – Question 41

Engineering Mathematics · Calculus and vector calculus · 2 marks · Multiple choice

Let $\boldsymbol\varphi=xy\hat i+yz\hat j+xz\hat k$. The closed surface $S$ is bounded by $x=0,y=0,z=0,x=3,y=2,z=1$, and $\hat n$ is its outward unit normal. The flux $\iint_S\boldsymbol\varphi\cdot\hat n\,dS$ is

  1. 18
  2. 9
  3. 36
  4. 3

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Correct answer: (A) 18

Explanation

The divergence is $\nabla\cdot\boldsymbol\varphi=y+z+x$. The divergence theorem gives its volume integral over the box. Volume is 6 and mean coordinates are 1.5, 1, 0.5. Flux is $6(1.5+1+0.5)=18$, **A**, with outward normals.