Line Integrals

A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics (for example, $W=F·s$) have natural continuous analogs in terms of line integrals ($W= \int_C F\cdot ds$). The line integral finds the work done on an object moving through an electric or gravitational field, for example.

In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve. More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by $z = f(x,y)$ and a curve $C$ in the $xy$-plane. The line integral of $f$ would be the area of the "curtain" created when the points of the surface that are directly over $C$ are carved out.

Line Integral Over Scalar Field

The line integral over a scalar field $f$ can be thought of as the area under the curve $C$ along a surface $z = f(x,y)$, described by the field.

Line Integral of a Scalar Field

For some scalar field $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:

$\int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt$

where $r: [a, b] \to C$ is an arbitrary bijective parametrization of the curve $C$ such that $r(a)$ and $r(b)$ give the endpoints of $C$ and $a$.

Line Integral of a Vector Field

For a vector field $\mathbf{F} : U \subseteq R^n \to R^n$, the line integral along a piecewise smooth curve $C \subset U$, in the direction of $r$, is defined as:

 $\displaystyle{\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt}$

where $\cdot$ is the dot product and $r: [a, b] \to C$ is a bijective parametrization of the curve $C$ such that $r(a)$ and $r(b)$ give the endpoints of $C$.