Given be a smooth manifold \(M\), a curve segment \(\gamma\colon [a,b]\to M\) and a smooth covector field \(\omega\in \mathfrak{X}^*(M)\). We define the line integral of \(\omega\) over \(\gamma\) by
\begin{equation*} \int_{\gamma} \omega = \int_{[a,b]} \gamma^*\omega = \int_{a}^{b} \omega_{\gamma(t)}\bigl(\gamma'(t)\bigr)dt , \end{equation*}where \(\gamma^*\omega \in \mathfrak{X}^*([a,b])\) denotes the pullback of \(\omega\) and is smooth function from \([a,b]\) to \(\mathbb{R}\).