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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 (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.    Next: Surface Integrals Up: Vectors and Vector Fields Previous: Line Integrals
Richard Fitzpatrick 2016-03-31