Line integral

A line integral of a vector function f along a line or curve segment C is the following (all three formulations are equivalent):

<math>\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}</math>

<math>\int_\mathbf{C} f_1s_1 dx + f_2s_2 dy + f_3s_3 dz</math>

<math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math>

Example: The work done on a particle to move it from one point to another is the line integral of the force on the particle along the curve of its motion:

<math>\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}</math>

Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero.

A line integral of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the line.