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SSC Pure Mathematics QUESTION #11136
Question 1
If $S$ is a closed surface and $v$ is the volume enclosed by $S$, then $\displaystyle\iint_S \mathbf{r}\cdot\mathbf{n}\,dS =$
  • $v$
  • $2v$
  • $3v$✔️
  • $4v$
Correct Answer Explanation
By the Divergence Theorem: $\displaystyle\iint_S \mathbf{r}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{r}\,dV$. Since $\mathbf{r} = x\,\mathbf{i}+y\,\mathbf{j}+z\,\mathbf{k}$, we have $\nabla\cdot\mathbf{r} = 1+1+1 = 3$. Therefore $\displaystyle\iint_S \mathbf{r}\cdot\mathbf{n}\,dS = \iiint_V 3\,dV = 3v$.