= Solution
Both paths start at $(0,0,0)$ and end at $(1,2,1)$. The potential values are $\phi(0,0,0)=C$ and $\phi(1,2,1)=4+C$, so
$$
\boxed{\Delta\phi=4=\int_{C_1}\mathbf F\cdot d\mathbf r=\int_{C_2}\mathbf F\cdot d\mathbf r}.
$$
This is the <fundamental theorem for line integrals>: an exact one-form integrates to the difference of its single-valued potential at the endpoints. The agreement is guaranteed for every piecewise smooth connecting path, rather than being an accident of these two examples. In particular the integral around a closed path assembled from one route and the reverse of the other is zero.
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