GATE 2020 MA – Question 61
Let $\mathbf F=(x+2y)e^z\hat i+(ye^z+x^2)\hat j+y^2z\hat k$. S is $x^2+y^2+z=1$, z ≥ 0, with either unit normal. If $|\iint_S(\nabla\times\mathbf F)\cdot\hat n\,dS|=\alpha\pi$, find α.
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Correct answer: 2 to 2
Explanation
Stokes’s theorem lets us replace the curved cap by its boundary circle and then by the unit disk at z = 0. There P = x + 2y and Q = y + x², so the planar curl is $Q_x-P_y=2x-2$. Its integral over the disk is $0-2\pi=-2\pi$, since the integral of x vanishes by symmetry. Taking absolute value gives 2π and α = 2.