Line integral (3-dimension)

Line integral of a scalar field

For some scalar field $f:U \subseteq \mathbb {R} ^{3}\rightarrow \mathbb {R}$, the line integral along a piecewise smooth curve ${\mathcal {C}}\subset \mathbb {U}$ is defined as

$$\int_{\mathcal{C}} f(\mathbf{r}) d s=\int_{a}^{b} f(\mathbf{r}(t))\left|\mathbf{r}^{\prime}(t)\right|\mathrm d t$$

$\blacksquare$