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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
. Line integrals of the form

(44) 
evaluated on some particular path taken between two fixed points and ,
often arise in Physics. Here
is
a path element. The path
might be specified as , , and , where
, , are mathematical functions, and is a parameter (such as pathlength) which varies monotonically
along the path. It follows that
.
In particular, if
is a forcefield then the line integral is the work done by the force in going between points
and along the given path [cf., Eq. (26)].
Finally, if the path is a closed loop (i.e., if and are the same point) then the integral is conventionally written

(45) 
As an example of a path integral, 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

(46) 
The second route is, firstly, around a large circle ( constant) to the point
(, , 0), and then parallel to the axissee Fig. 8. In the first, part no work is
done,
since is perpendicular to . In the second part,

(47) 
In this case, the integral is independent of the path taken between the beginning and end points. However, not all line integrals
are path independent. Indeed, there are two different classes of
line integralthose whose values only depend on the end points, and
those whose values depend both on the end points and the path taken
between these points.
Figure 8:
An example line integral.

Next: Surface Integrals
Up: Vectors
Previous: Vector Calculus
Richard Fitzpatrick
20070714