GATE 2019 EE – Question 49

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 · Numerical answer

If $\mathbf A=2x\mathbf i+3y\mathbf j+4z\mathbf k$ and $u=x^2+y^2+z^2$, then $\text{div}(u\mathbf A)$ at $(1,1,1)$ is ________.

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Correct answer: 45

Explanation

$\nabla\cdot(u\mathbf A)=u\,\nabla\cdot\mathbf A+\mathbf A\cdot\nabla u$. At $(1,1,1)$: $u=3$ and $\nabla\cdot\mathbf A=2+3+4=9$, so the first term is $27$. $\nabla u=(2,2,2)$ gives $\mathbf A\cdot\nabla u=2\cdot2+3\cdot2+4\cdot2=18$. The total is $45$.