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Figure 11:

Consider a twodimensional function which is defined for all and .
What is meant by the integral of along a given curve from to in the  plane?
We first draw out as a function of length along the path (see Fig. 11). The integral is then simply given
by

(60) 
As an example of this, consider the integral of between and along the
two routes indicated in Fig. 12.
Along route 1 we have , so
. Thus,

(61) 
The integration along route 2 gives
Note that the integral depends on the route taken between the initial and final points.
Figure 12:

The most common type of line integral is that where the contributions from and are evaluated
separately, rather that through the path length :

(63) 
As an example of this, consider the integral

(64) 
along the two routes indicated in Fig. 13.
Along route 1 we have and , so

(65) 
Along route 2,

(66) 
Again, the integral depends on the path of integration.
Figure 13:

Suppose that we have a line integral which does not depend on the path of integration. It
follows that

(67) 
for some function . Given for one point in the  plane, then

(68) 
defines for all other points in the plane. We can then draw a contour map of .
The line integral between points and is simply the change in height in the contour
map between these two points:

(69) 
Thus,

(70) 
For instance, if then
and

(71) 
is independent of the path of integration.
It is clear that there are two distinct types of line integral. Those which depend only on their
endpoints and not on the path of integration, and those which depend both on their endpoints
and the integration path. Later on, we shall learn how to distinguish between these two types.
Next: Vector line integrals
Up: Vectors
Previous: Vector calculus
Richard Fitzpatrick
20060202