GATE 2014 EC (EC4) – Question 64
$\mathbf F=z\hat a_x+x\hat a_y+y\hat a_z$. S is the part of $x^2+y^2+z^2=1$ with z ≥ 0. Find $\int_S(\nabla\times\mathbf F)\cdot d\mathbf S$.
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Correct answer: 3.13 to 3.15
Explanation
For F=(z,x,y), curlF=(1,1,1). With outward orientation, symmetry cancels the xand yfluxes through the hemisphere; the zflux equals its projected disk areaπ. Equivalently, apply Stokes’ theorem around the counterclockwise equator. Evaluate F at z=0:(0,cosθ,sinθ), dotted with dl=(−sinθ,cosθ,0)dθ, giving∫₀²πcos²θdθ=**π**.