Let's say i have the solid $E$ bounded by the cylinder $x^2+y^2=2x$ and the upper part of the sphere $x^2+y^2+z^2=4$. Also i have the vector field $F=(y-z, yz, -xz)$.
I was asked to find the circulation of $rot \ F$ through the vertical and spherical face of $E$.
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I know i have to use Stokes theorem (or perhaps i can work with Gauss theorem). The real problem is how do i work with this solid?
$\nabla\cdot\nabla \times F = 0$
So using Gauss' rule. The net flow is 0.
sphere + cylinder wall + disk = 0
$\int \nabla \times F = \oint F \cdot dr$
Through the bottom disk.
$r = (\cos \theta+1, \sin \theta, 0)$
$\int_0^{2\pi} -\sin^2\theta d\theta = \pi$
But that is oriented clockwise.
If we flip the orientation (pointing outward, instead of upward) $-\pi$
Through the top of the sphere:
$r = (\cos \theta+1, \sin \theta, \sqrt {2-2cos\theta})\\ \sqrt {2-2cos\theta} = 2\sin \frac 12 \theta$
$\int_0^{2\pi} -\sin^2\theta + (2-2\cos\theta)(-\sin\theta) + (\sin\theta)(2-\cos\theta)(\cos\theta)+ (1+\cos\theta)(\sin\frac12\theta)(2\sin\theta) d\theta$