Line integral
In mathematics, a line integral is an integral where a scalar or vector function to be integrated is along a curve \(C\). Its value can depend on the path of integration.
Scalar functions
The line integral of \(\require{physics} f( \vb{r} ) = f(x,y,z)\) along curve \(C\) is denoted by \[\begin{equation} \require{physics} \int_C f( \vb{r} ) \dd{s} % \thinspace , \end{equation}\] where \(\dd{s}\) is interpreted as an arc length and is here to remind us that we are moving along curve \(C\) instead of an axis. To move along \(C\), we can parametrize \(x\), \(y\) and \(z\) using \(t\): \[\begin{equation} \int_C f( \vb{r} ) \dd{s} % = \int_a^b f( \vb{r}(t) ) \dd{s} % \thinspace , \end{equation}\] where we are still stuck with the parametrization of \(\dd{s}\). Arc length is parametrized by \[\begin{equation} \dd{s} % = \sqrt{ \qty( \pdv{x}{t} )^2 + \qty( \pdv{y}{t} )^2 + \qty( \pdv{z}{t} )^2 } \dd{t} \thinspace , \end{equation}\] resulting in \[\begin{align} \int_C f( \vb{r} ) \dd{s} % &= \int_a^b f( \vb{r}(t) ) \sqrt{ \qty( \pdv{x}{t} )^2 + \qty( \pdv{y}{t} )^2 + \qty( \pdv{z}{t} )^2 } \dd{t} \\ &= \int_a^b f( \vb{r}(t) ) \norm{ f( \vb{r}'(t) ) } \dd{t} \end{align}\] for the line integral.