Vector Line Integrals

Consider a general vector field . Let be the vector element of line length. Vector line integrals often arise as

(A.81) |

For instance, if is a force-field then the line integral is the work done in going from to .

For example, consider the work done by a repulsive inverse-square central field, . The element of work done is . Take and . The first route considered is along the -axis, so

(A.82) |

The second route is, firstly, around a large circle ( constant) to the point ( , , 0), and then parallel to the -axis. (See Figure A.16). In the first part, no work is done, because is perpendicular to . In the second part,

(A.83) |

In this case, the integral is independent of the path. However, not all vector line integrals are path independent.