Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics V / 2 / a / i / Solution

For a continuous vector field along an oriented piecewise continuously differentiable parametric curve $\gamma :[a,b]\to {\mathbb{R}}^3$, define its line integral byadding the integrals over the smooth pieces when necessary. The value refers to the specified traversal: going around a curve twice counts twice.