Line integral of a scalar field

Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a rectifiable curve with image $C$, and let $f \colon C \to \mathbb{R}$ be continuous. For a partition $a = t_0 < t_1 < \dots < t_k = b$ and points $\tau_i \in [t_{i-1}, t_i]$, let $\Delta s_i$ be the length of the restriction of $\gamma$ to $[t_{i-1}, t_i]$. The line integral of $f$ along $\gamma$ is the limit $$\int_\gamma f \, ds = \lim \sum_{i=1}^{k} f(\gamma(\tau_i)) \, \Delta s_i,$$ taken as $\max_i (t_i - t_{i-1}) \to 0$.