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## Vector line integrals

A vector field is defined as a set of vectors associated with each point in space. For instance, the velocity in a moving liquid (e.g., a whirlpool) constitutes a vector field. By analogy, a scalar field is a set of scalars associated with each point in space. An example of a scalar field is the temperature distribution in a furnace.

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

 (72)

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

As an example, consider the work done in a repulsive, inverse-square, central field, . The element of work done is . Take and . Route 1 is along the -axis, so

 (73)

The second route is, firstly, around a large circle ( constant) to the point (, , 0), and then parallel to the -axis. In the first, part no work is done, since is perpendicular to . In the second part,
 (74)

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

Next: Surface integrals Up: Vectors Previous: Line integrals
Richard Fitzpatrick 2006-02-02