Line integral

A line integral of a vector function f along a line or curve segment C is the following (all three formulations are equivalent):

$\int _{\mathbf {C} }{\vec {f}}({\vec {s}})\cdot d{\vec {s}}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\mathbf{C} f_1s_1 dx + f_2s_2 dy + f_3s_3 dz}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle}

Example: The work done on a particle to move it from one point to another is the line integral of the force on the particle along the curve of its motion:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}}

A line integral of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the line.