Question

Let m be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x2+y2=49, 0≤z≤1, and a hemispherical cap defined by x2+y2+(z−1)2=49, z≥1. for the vector field f=(zx+z2y+8y, z3yx+5x, z4x2), compute ∬m(∇×f)⋅ds in any way you like.

Answer 1

It seems that the boundary of $M$ is the circle $x^2+y^2=49$ in the plane $z=0$. By Stokes' theorem,

$\displaystyle\iint_M(\nabla\times\vec f)\cdot\mathrm d\vec S=\int_{\partial M}\vec f\cdot\mathrm d\vec r$

Parameterize $\partial M$ by

$\vec r(t)=(7\cos t,7\sin t,0)$

with $0\le t\le2\pi$. Then the line integral is

$\displaystyle\int_{\partial M}\vec f(x(t),y(t),z(t))\cdot\mathrm d\vec r=\int_0^{2\pi}(56\sin t,35\cos t,0)\cdot(-7\sin t,7\cos t,0)\,\mathrm dt$

$=\displaystyle\int_0^{2\pi}(245\cos^2t-392\sin^2t)\,\mathrm dt=\boxed{-147\pi}$