Summary of Line Integrals

Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the x-axis, but the domain of integration in a line integral is a curve in a plane or in space.
If $C$ is a curve, then the length of $C$ is $\displaystyle\int_{C} ds$ .
There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.
Scalar line integrals can be calculated using $\displaystyle\int_{C} f(x,y,z)ds=\displaystyle\int_{a}^{b} f({\bf{r}}(t))\sqrt{(x^{\prime}(t))^{2}+(y^{\prime}(t))^{2}+(z^{\prime}(t))^{2}}dt$ ; vector line integrals can be calculated using $\displaystyle\int_{C} {\bf{F}}\cdot{ds}=\displaystyle\int_{C}{\bf{F}}\cdot{\bf{T}}ds=\displaystyle\int_{a}^{b}{\bf{F}}({\bf{r}}(t))\cdot{\bf{r}}^{\prime}(t)dt$ .
Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.

Key Equations

Calculating a scalar line integral
$\displaystyle\int_{C} f(x,y,z)ds=\displaystyle\int_{a}^{b} f({\bf{r}}(t))\sqrt{(x^{\prime}(t))^{2}+(y^{\prime}(t))^{2}+(z^{\prime}(t))^{2}}dt$
Calculating a vector line integral
$\displaystyle\int_{C} {\bf{F}}\cdot{ds}=\displaystyle\int_{C}{\bf{F}}\cdot{\bf{T}}ds=\displaystyle\int_{a}^{b}{\bf{F}}({\bf{r}}(t))\cdot{\bf{r}}^{\prime}(t)dt$
or
${\displaystyle\int_{C}} Pdx+Qdy+Rdz= {\displaystyle\int_{a}^{b}}\left(P({\bf{r}}(t))\frac{dx}{dt}+Q({\bf{r}}(t))\frac{dy}{dt}+R({\bf{r}}(t))\frac{dz}{dt}\right)dt$
Calculating flux
$\displaystyle\int_{C} {\bf{F}}\cdot{\frac{{\bf{n}}(t)}{\Arrowvert{\bf{n}}(t)\Arrowvert}}ds=\displaystyle\int_{a}^{b}{\bf{F}}({\bf{r}}(t))\cdot{\bf{n}}(t)dt$

circulation: the tendency of a fluid to move in the direction of curve $C$ . If $C$ is a closed curve, then the circulation of ${\bf{F}}$ along $C$ is line integral $\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds$ , which we also denote $\displaystyle\oint_{C} {\bf{F}}\cdot{\bf{T}}ds$

flux: the rate of a fluid flowing across a curve in a vector field; the flux of vector field ${\bf{F}}$ across plane curve $C$ is line integral $\displaystyle\int_{C} {\bf{F}}\cdot{\frac{{\bf{n}}(t)}{\Arrowvert{\bf{n}}(t)\Arrowvert}}ds$

line integral: the integral of a function along a curve in a plane or in space

orientation of a curve: the orientation of a curve $C$ is a specified direction of $C$

piecewise smooth curve: an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves

scalar line integral: the scalar line integral of a function $f$ along a curve $C$ with respect to arc length is the integral $\displaystyle\int_C \! f\, \mathrm{d}s$ , it is the integral of a scalar function $f$ along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral

vector line integral: the vector line integral of vector field ${\bf{F}}$ along curve $C$ is the integral of the dot product of ${\bf{F}}$ with unit tangent vector ${\bf{T}}$ of $C$ with respect to arc length, $\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds$ ; such an integral is defined in terms of a Riemann sum, similar to a single-variable integral