> For the complete documentation index, see [llms.txt](https://coldmoon.gitbook.io/mathematical-analysis/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://coldmoon.gitbook.io/mathematical-analysis/mathematical-analysis-2-1/8.-integral/1-dimension-curves.md).

# 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

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

$$\blacksquare$$