---- > [!definition] Definition. ([[line integral]]) > Let $\gamma=\boldsymbol r:I=[a,b] \to \mathbb{R}^{n}$ be a regular [[parameterized curve]], so $C:=\gamma(I)$ is a differentiable [[smooth immersion|immersed]] $1$-[[differentiable Euclidean submanifold|sub]][[smooth manifold|manifold]] of $\mathbb{R}^{n}$ [[coordinate chart|charted by]] $t:C \to [a,b]$. Suppose $f: C \to \mathbb{R}$ is scalar function/field on $C$. > In this case the canonical [[differential form|volume form]] $\mathrm{d}V=ds$ equals $\|\gamma'\| dt$, and so [[integration of a compactly supported volume form on an oriented smooth manifold|the]] [[single-patch scalar integral over a compact manifold|scalar integral]] $\int _{C} f \ \mathrm{d}V$ is $\int _{C} f \ \mathrm{d} s = \int _{I} (f \circ \boldsymbol r) \|\gamma'\| = \int _{a}^{b} f\big( \boldsymbol r(t) \big) \|\gamma'(t)\| \ dt,$where the middle is ordinary 'careful notation' [[Riemann integral]] and the RHS is the [[Riemann integral]] with classical calculus notation. We call $\int _{C} f \ \d s$ the **(scalar) line integral of $f$ over $C$ (with respect to [[arc length of a path|arc length]] $s$)**. > There is also a notion of 'line integral of a [[vector field]]': see [[circulation integral]]. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```