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GATE 2026 EE – Question 32

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 1 mark · Numerical answer

Given that $\vec F(x,y,z)=\sin(y)\,\hat x+\cos(x)\,\hat y+5\,\hat z$, the integral $\oiint_S\vec F(x,y,z)\cdot d\vec s$ over the unit sphere $S$ centered at the origin evaluates to ______.

(Round off to one decimal place)

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Correct answer: -0.01 to 0.01

Explanation

By the divergence theorem the surface integral equals $\iiint\nabla\cdot\vec F\,dV$. $\nabla\cdot\vec F=\dfrac{\partial}{\partial x}\sin y+\dfrac{\partial}{\partial y}\cos x+\dfrac{\partial}{\partial z}5=0$. So the integral is 0.0.