= Solution
The <curl> is constant:
$$
\nabla\times\mathbf F=(\partial_yF_z-\partial_zF_y,\partial_zF_x-\partial_xF_z,\partial_xF_y-\partial_yF_x)=(1,1,1).
$$
Orient the spanning planar disk by the <unit normal> $\mathbf n=(a,b,c)$ and orient its boundary by the right-hand rule, equivalently counterclockwise when viewed from the side towards which $\mathbf n$ points. <Stokes theorem> converts the <line integral> to a constant <flux integral> through a disk of area $\pi R^2$. \b[With this orientation]
$$
\boxed{\oint_C\mathbf F\cdot d\mathbf x=\pi R^2(a+b+c).}
$$
Reversing the orientation changes the sign. The centre of the circle does not enter because the <curl> is constant.
Back to article page
