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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

 (1338)

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

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

 (1339)

The second route is, firstly, around a large circle ( constant) to the point (, , 0), and then parallel to the -axis--see Figure A.110. In the first part, no work is done, since is perpendicular to . In the second part,
 (1340)

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

Next: Volume Integrals Up: Vector Algebra and Vector Previous: Line Integrals
Richard Fitzpatrick 2011-03-31