327 Line Integral

A scalar obtained by integrating a field along a curve that is used to measure accumulated value along a path.

definition [d] (Line Integral = Path Integral = Curve Integral) The scalar obtained by integrating a continuous vector field $\mathbf{F}$ along a smooth curve $C$ given by $\mathbf{r}(t)$, $a \leq t \leq b$:

$\displaystyle \int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{a}^{b} \mathbf{F}\!\bigl(\mathbf{r}(t)\bigr) \cdot \dfrac{d\mathbf{r}}{dt}(t)\, dt$ .

where

$C$ is the path of integration.
$\mathbf{F}$ is a vector field.
$\mathbf{r}(t)$ is a parametrization of $C$.
$t$ is the parameter.
$a$ and $b$ are the parameter endpoints.
$d\mathbf{r} = \dfrac{d\mathbf{r}}{dt}(t)\, dt$ is the infinitesimal displacement along $C$.
$\dfrac{d\mathbf{r}}{dt}(t)$ is the tangent vector along $C$.

$Vector field \mathbf{F} along a smooth curve C from A to B. At a point P on C, the field \mathbf{F} and the differential displacement d\mathbf{r} are shown in the direction of integration.$

Vector field $\mathbf{F}$ along a smooth curve $C$ from $A$ to $B$. At a point $P$ on $C$, the field $\mathbf{F}$ and the differential displacement $d\mathbf{r}$ are shown in the direction of integration.

The Riemann-sum breakdown of the line integral is

\[ \int_{C} \mathbf{F}\cdot d\mathbf{r} = \lim_{n\to\infty} \sum_{i=1}^{n} \mathbf{F}(\mathbf{r}_{i})\cdot\Delta\mathbf{r}_{i} \]

where

$C$ is the path of integration.
$\mathbf{F}$ is a vector field.
$n$ is the number of segments of $C$.
$\mathbf{r}_{i}$ is a sample point on the $i$th segment.
$\Delta\mathbf{r}_{i}$ is the displacement along the $i$th segment.

definition [d] (Line Integral = Path Integral) The scalar obtained by integrating a continuous scalar function $f$ with respect to arc length along a smooth curve $C$ given by $\mathbf{r}(t)$, $a \leq t \leq b$:

$\displaystyle \int_{C} f\, ds = \int_{a}^{b} f\!\bigl(\mathbf{r}(t)\bigr)\, \left\lvert \dfrac{d\mathbf{r}}{dt}(t) \right\rvert\, dt$ .

where

$C$ is the path of integration.
$f$ is a scalar function.
$\mathbf{r}(t)$ is a parametrization of $C$.
$t$ is the parameter.
$a$ and $b$ are the parameter endpoints.
$ds = \left\lvert \dfrac{d\mathbf{r}}{dt}(t) \right\rvert\, dt$ is the arc-length element.
$\dfrac{d\mathbf{r}}{dt}(t)$ is the tangent vector along $C$.

327.1 References

Stewart, J. Calculus. — arc length $L = \int_{a}^{b} \left\lvert \dfrac{d\mathbf{r}}{dt}(t) \right\rvert\, dt$; line / path integrals (closest indexed material; dedicated line-integral chapter not fully excerpted in the notebook).
Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011. — line / curve integral.
Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — path integral / work integral.

326 Limit

328 Notes