> [!theorem] > > Let $X$ be a [[Smooth Manifold|smooth manifold]] and $f \in C^\infty(X)$ be a [[Function on Manifold|smooth function]]. Let $\gamma: [a, b] \to X$ be a piecewise smooth [[Curve|curve]] segment, then > $ > \int_\gamma df = f(\gamma(b)) - f(\gamma(a)) > $ > *Proof*. By the ordinary fundamental theorem of calculus, > $ > \begin{align*} > \int_\gamma df &= \int_{[a, b]} \angles{df_{\gamma(t)}, d\gamma_{t}} \\ > &= \int_{[a, b]}(f \circ \gamma)'(t)dt = f(\gamma(b)) - f(\gamma(a)) > \end{align*} > $