Both paths start at

(0, 0, 0)

and end at

(1, 2, 1)

. The potential values are

ϕ (0, 0, 0) = C

and

ϕ (1, 2, 1) = 4 + C

, so

Δ ϕ = 4 = \int_{C_{1}} F \cdot d r = \int_{C_{2}} F \cdot d r .

(1)

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.