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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, inversesquare,
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
20060202