Table 9.5.1   Salient points relevant to line integrals

In 2-D

In 3-D

Explicitly: $y\=y\left(x\right)$

Parametrically: $x\=x\left(t\right)\,y\=y\left(t\right)$

Parametrically, as a position vector: $\mathbf{R}\=\left[\begin{array}{c}x\(t\)\\ y\(t\)\end{array}\right]$

Parametrically: $x\=x\left(t\right)\,y\=y\left(t\right)\,z\=z\left(t\right)$

Parametrically, as a position vector: $\mathbf{R}\=\left[\begin{array}{c}x\(t\)\\ y\(t\)\\ z\(t\)\end{array}\right]$

Table 9.5.2   Representing paths of integration for line integrals

Representation of $C$

Implementation of the line integral ${\int }_{C}f \mathrm{ds}$

$y\=y\left(x\right)$

${\int }_{x\=a}^{x\=b}f\left(x\,y\left(x\right)\right) \sqrt{1\+{\left(y\prime \right)}^2} \mathrm{dx}$

$x\=x\left(t\right)\,y\=y\left(t\right)$

$\mathbf{R}\=\left[\begin{array}{c}x\(t\)\\ y\(t\)\end{array}\right]$

${\int }_{t\=a}^{t\=b}f\left(x\left(t\right)\,y\left(t\right)\right) \sqrt{{\stackrel{\.}{x}}^2\+{\stackrel{\.}{y}}^2} \mathrm{dt}$

$x\=x\left(t\right)\,y\=y\left(t\right)\,z\=z\left(t\right)$

$\mathbf{R}\=\left[\begin{array}{c}x\(t\)\\ y\(t\)\\ z\(t\)\end{array}\right]$

${\int }_{t\=a}^{t\=b}f\left(x\left(t\right)\,y\left(t\right)\,z\left(t\right)\right) \sqrt{{\stackrel{\.}{x}}^2\+{\stackrel{\.}{y}}^2\+{\stackrel{\.}{z}}^2} \mathrm{dt}$

Table 9.5.3   Line integrals of scalar fields

The line integral of the tangential component of a vector field $\mathbf{F}\=f \mathbf{i}\+g \mathbf{j}\+h \mathbf{k}$ will typically be represented by any of the following three notations

${\int }_C\mathbf{F}·\mathbf{dr}$ = ${\int }_C\mathbf{F}·\mathbf{T} \mathrm{ds}$ = ${\int }_{C}f \mathrm{dx}\+g \mathrm{dy}\+h \mathrm{dz}$ 

where $C$ is the path for the integration, $s$ is arc length along that path, T is a unit tangent vector along $C$, and r is a position-vector representation of the parametrically given $C$. Some texts will write dr for the differential of the vector r, and some will write dr. In either event, the symbol represents the differential of the vector $\mathbf{r}\left(t\right)$, so it is a derivative times an increment: $\mathbf{dr}\=\stackrel{\.}{\mathbf{r}}\left(t\right)\mathrm{dt}$ = $\left(\stackrel{\.}{x}\left(t\right) \mathbf{i}\+\stackrel{\.}{y}\left(t\right) \mathbf{j}\+\stackrel{\.}{z}\left(t\right) \mathbf{k}\right)\mathrm{dt}$ = $\mathrm{dx} \mathbf{i}\+\mathrm{dy} \mathbf{j}\+\mathrm{dz} \mathbf{k}$.

If the curve $C$ is parametrized by arc length, then $\mathbf{r}\prime \left(s\right)\=\mathbf{T}\left(s\right)$ is automatically a unit tangent vector, and $\mathbf{dr}\=\mathbf{T} \mathrm{ds}$. It should now be clear why the three forms of the line integral of the tangential component of F are equivalent.

Table 9.5.4   Line integral of the tangential component of F

Flux of the field $\mathbf{F}\=f\left(x\,y\right) \mathbf{i}\+g\left(x\,y\right) \mathbf{j}$ through the plane curve $C$, given by $\mathbf{r}\=x\left(t\right) \mathbf{i}\+y\left(t\right) \mathbf{j}$, is represented by the line integral of the normal component of F along C, for which either of the following two notations suffice.

${\int }_C\mathbf{F}·\mathbf{N} \mathrm{ds}$ = ${\int }_{C}f \mathrm{dy}-g \mathrm{dx}$ 

Indeed, if r is parametrized by arc length $s$, then $\mathbf{T}\=x\prime \left(s\right) \mathbf{i}\+y\prime \left(s\right) \mathbf{j}$ is a unit tangent vector along C, and a normal vector to the right of T is $\mathbf{N}\=y\prime \left(s\right) \mathbf{i}-x\prime \left(s\right) \mathbf{j}$. Consequently, $\mathbf{F}·\mathbf{N} \mathrm{ds}\=f \mathrm{dy}-g \mathrm{dx}$.

A mnemonic for remembering the flux integral is the word "flux" itself. Its first letter is the first letter of the integrand; its last, the last letter of the integrand. These two letters indicate where the function $f\left(x\,y\right)$ and the differential $\mathrm{dx}$ go; the function $g\left(x\,y\right)$ and the differential $\mathrm{dy}$ have to fall into place on either side of the minus sign.

Finally, note that for the line integral of the normal component of F, there is no analog for the $\mathbf{F}·\mathbf{dr}$ appearing in the line integral of the tangential component.

Table 9.5.5   Flux through a plane curve, the line integral of the normal component of a field F.

Scalar

Tangential Component

Normal Component

Student: PathInt

Main:      PathInt

Student: LineInt

Main:      LineInt

Student: Flux

Main:      Flux

Arc

Ellipse

Circle

Line

LineSegments

Path

Arc

Ellipse

Circle

Line

LineSegments

Path

Arc

Ellipse

Circle

Line

LineSegments

Path

Table 9.5.6   Commands for line integrals

Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻

Line Integrals

Flux

 (There are seven others for flux through surface.)

Table 9.5.7   Task templates for line integrals

Example 9.5.1

Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken along the line segment from $\left(1\,2\right)$ to $\left(3\,5\right)$.

Example 9.5.2

Obtain the line integral of the scalar function $f\left(x\,y\,z\right)\=x y z$, taken along the line segment from $\left(1\,2\,3\right)$ to $\left(5\,3\,2\right)$.

Example 9.5.3

Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken along the polygonal line connecting the points $\left(1\,2\right)$, $\left(3\,5\right)$, $\left(4\,0\right)$, $\left(2\,7\right)$, in that order.

Example 9.5.4

Obtain the line integral of the scalar function $f\left(x\,y\,z\right)\=x y z$, taken along the polygonal line connecting the points $\left(1\,2\,3\right)$, $\left(5\,3\,2\right)$, and $\left(0\,1\,4\right)$, in that order.

Example 9.5.5

Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken around the circle whose center is $\left(3\,5\right)$ and whose radius is 2.

Example 9.5.6

Let $C$ be the circle whose center is $\left(3\,5\right)$ and whose radius is 2. Let $c$ be the arc on $C$ subtended by a central angle of $\mathrm{\π}\/4$ radians measured counterclockwise from the right half of a horizontal diagonal. Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken along $c$.

Example 9.5.7

Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken around the ellipse whose center is $\left(3\,4\right)$ and whose semi-major and semi-minor axes, parallel to the coordinate axes, are 2 and 1, respectively.

Example 9.5.8

Let $C$ be the ellipse $x^2\+4 y^2\=1$ and let $c$ be the arc subtended by an angle of $\mathrm{\π}\/4$ radians measured counterclockwise from the positive $x$-axis. Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken along $c$.

Example 9.5.9

Let $C$ be that portion of the parabola $y\=x^2$ from the origin to the point $\left(1\,1\right)$. Obtain the line integral of the scalar function $f\left(x\,y\right)\=x y$, taken along $C$.

Example 9.5.10

Calculate the work that the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ does on a unit positive charge as the charge moves along a line from $\left(1\,2\right)$ to $\left(3\,5\right)$.

Example 9.5.11

Calculate the work that the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ does on a unit positive charge as the charge moves along the polygonal line connecting the points $\left(1\,2\right)$, $\left(3\,5\right)$, $\left(4\,0\right)$, $\left(2\,7\right)$, in that order.

Example 9.5.12

Calculate the circulation of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ on the circle whose center is $\left(3\,5\right)$ and whose radius is 2.

Example 9.5.13

Let $C$ be the circle whose center is $\left(3\,5\right)$ and whose radius is 2. Let $c$ be the arc on $C$ subtended by a central angle of $\mathrm{\π}\/4$ radians measured counterclockwise from the right half of a horizontal diagonal. Calculate the work that the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ does on a unit positive charge as the charge moves counterclockwise along $c$.

Example 9.5.14

Calculate the circulation of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ on the ellipse whose center is $\left(3\,4\right)$ and whose semi-major and semi-minor axes, parallel to the coordinate axes, are 2 and 1, respectively.

Example 9.5.15

Let $C$ be the ellipse $x^2\+2 y^2\=1$ and let $c$ be the arc subtended by an angle of $\mathrm{\π}\/4$ radians measured counterclockwise from the positive $x$-axis. Calculate the work that the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ does on a unit positive charge as the charge moves counterclockwise along $c$.

Example 9.5.16

Let $C$ be that portion of the parabola $y\=x^2$ from the origin to the point $\left(1\,1\right)$. Calculate the work that the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ does on a unit positive charge as the charge moves away from the origin along $C$.

Example 9.5.17

Calculate the flux of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ through the line from $\left(1\,2\right)$ to $\left(3\,5\right)$. Take the normal to the right of the tangent along the line.

Example 9.5.18

Calculate the flux of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ through the polygonal line connecting the points $\left(1\,2\right)$, $\left(3\,5\right)$, $\mathrm{OmegaR}≔\left[\mathrm{dx1}\,{\ⅇ}^{\mathrm{x1}}\left(\mathrm{x3}\+\mathrm{x1}\mathrm{x3}\+\mathrm{x2}\right)\mathrm{dx1}\+\mathrm{dx2}\,\mathrm{x3}{\ⅇ}^{\mathrm{x1}}\mathrm{dx1}\+\mathrm{dx3}\right]$, $\left(7\,6\right)$, in that order. Take the normal to the right of the tangent along the path.

Example 9.5.19

Calculate the flux of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ through the circle whose center is $\left(3\,5\right)$ and whose radius is 2. Take the outward normal along the circle.

Example 9.5.20

Calculate the flux of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ through the ellipse whose center is $\left(3\,4\right)$ and whose semi-major and semi-minor axes, parallel to the coordinate axes, are 2 and 1, respectively. Take the outward normal along the ellipse.

Example 9.5.21

Let $C$ be that portion of the parabola $y\=x^2$ from the origin to the point $\left(1\,1\right)$. Calculate the flux of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ through $C$. Take the normal to the right of the tangent along the path.

Example 9.5.22

Let $C$ be the circle whose center is $\left(3\,5\right)$ and whose radius is 2. Let $c$ be the arc on $C$ subtended by a central angle of $\mathrm{\π}\/4$ radians measured counterclockwise from the right half of a horizontal diagonal. Calculate the flux of the field $\mathbf{F}\=x y \mathbf{i}\+\left(x\+y\right) \mathbf{j}$ through $c$. Take the normal to the right of the tangent along $c$.