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 [latex]C[/latex] is a curve, then the length of [latex]C[/latex] is [latex]\displaystyle\int_{C} ds[/latex].
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[latex]\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[/latex]; vector line integrals can be calculated using [latex]\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[/latex].
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
[latex]\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[/latex]
Calculating a vector line integral
[latex]\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[/latex]
or
[latex]{\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[/latex]
Calculating flux
[latex]\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[/latex]

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

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

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 [latex]C[/latex] is a specified direction of [latex]C[/latex]

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 [latex]f[/latex] along a curve [latex]C[/latex] with respect to arc length is the integral [latex]\displaystyle\int_C \! f\, \mathrm{d}s[/latex], it is the integral of a scalar function [latex]f[/latex] 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 [latex]{\bf{F}}[/latex] along curve [latex]C[/latex] is the integral of the dot product of [latex]{\bf{F}}[/latex] with unit tangent vector [latex]{\bf{T}}[/latex] of [latex]C[/latex] with respect to arc length, [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex]; such an integral is defined in terms of a Riemann sum, similar to a single-variable integral