# Line Integrals on Intervals > [!definition] > > Let $[a, b] \subset \real$ be a closed interval, and $\omega$ be a $C^p$ [[Covector Field|covector field]] along $[a, b]$. If $t$ is the standard coordinate on $\real$, then there exists $f \in C^p([a, b])$ such that $\omega_t = f(t)dt$. The **integral** of $\omega$ over $[a, b]$ is > $ > \int_{[a, b]}\omega = \int_{[a, b]}f(t)dt > $ > [!theorem] Diffeomorphism Invariance of Line Integral > > Let $[a, b], [c, d] \subset \real$ be closed intervals, $\omega$ be a $C^1$ covector field along $[a, b]$, and $\varphi: [c, d] \to [a, b]$ be an increasing $C^1$-isomorphism. Then > $ > \int_{[c, d]}\varphi^*\omega = \int_{[a, b]}\omega > $ > If $\varphi$ is decreasing instead, then $\int_{[c, d]}\varphi^*\omega = -\int_{[a, b]}\omega$. > > *Proof*. Let $s$ and $t$ be standard coordinates on $[c, d]$ and $[a, b]$, respectively. Let $f: [a, b] \to \real$ be such that $\omega = f(t)dt$, then the [[Pullback of Covector Field|pullback]] has coordinate expression > $ > \varphi^*\omega = \varphi^{*}(f \cdot dt) = (f \circ \varphi) \cdot (\varphi' ds) > $ > This exactly corresponds to a change of variables formula for ordinary integrals, hence > $ > \int_{[c, d]}\varphi^*\omega = \int_{[c, d]}(f \circ \varphi)(s) \cdot \abs{\varphi'(s)}ds = \int_{[a, b]}f(t)dt = \int_{[a, b]}\omega > $ # Line Integrals on Manifolds > [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] ($p \ge 2$), $\gamma: [a, b] \to X$ be a $C^{1}$-curve segment, and $\omega \in \vf^*(X)$ be a $C^{1}$-[[Covector Field|covector field]]. The **line integral** of $\omega$ over $\gamma$ is > $ > \int_\gamma \omega = \int_{[a, b]}\gamma^*\omega > $ > If $\gamma$ is piecewise $C^{p - 1}$ with the partition $\seqf{a_i}$ with $a_0 = a$, then > $ > \int_\gamma \omega = \sum_{i = 1}^k\int_{[a_{i - 1}, a_i]}\gamma^*\omega > $ > [!theorem] > > Let $X, Y$ be $C^p$-manifolds ($p \ge 2$), $\gamma: [a, b] \to X$. If $\omega \in \vf^*(Y)$ and $F: M \to N$ be a $C^p$-morphism, then > $ > \int_\gamma F^* \omega = \int_{F \circ \gamma}\omega > $ > [!theorem] > > Let $\gamma: [a, b] \to X$ and $\eta: [c, d] \to X$ be two $C^1$ curve segments. If there exists a $C^1$ [[Diffeomorphism|diffeomorphism]] $\varphi: [c, d] \to [a, b]$ such that $\eta = \gamma \circ \varphi$, then $\eta$ is a **reparametrisation** of $\gamma$. > > If $\varphi$ is increasing, then $\eta$ is an *increasing* reparametrisation, and > $ > \int_{\eta} \omega = \int_{\gamma}\omega > $ > If $\varphi$ is decreasing, then $\eta$ is a *decreasing* reparametrisation, and > $ > \int_\eta \omega = -\int_\gamma \omega > $ > [!theorem] > > Let $\gamma: [a, b] \to X$ be a piecewise smooth curve segment, then > $ > \int_\gamma \omega = \int_{[a, b]}\angles{\omega_{\gamma(t)}, d\gamma_t}dt > $ > *Proof*. > $ > \int_\gamma \omega = \int_{[a, b]}\gamma^*\omega = \int_{[a, b]}\angles{d\gamma_t^*, \omega_{\gamma(t)}} > $ > where > $ > \angles{d\gamma_t^*, \omega_{\gamma(t)}} = \angles{\omega_{\gamma(t)}, d\gamma_t}dt > $