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GATE 2025 EE – Question 51

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 2 marks · Multiple select

Let $\boldsymbol a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $\boldsymbol f(\boldsymbol R)=\boldsymbol a_R\dfrac1{R^n}$ is independent of $R$?

  1. −2
  2. −1
  3. 1
  4. 2

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Correct answer: (B) −1; (D) 2

Explanation

$\nabla\cdot\boldsymbol f=\dfrac1{R^2}\dfrac{d}{dR}\left(R^2R^{-n}\right)=(2-n)R^{-n-1}$. It is independent of $R$ when $n=-1$ (a constant 3) or when the coefficient vanishes, $n=2$ (identically zero). For $n=-2$ it is $4R$ and for $n=1$ it is $1/R^2$.