The line integral was then precisely defined in terms of a limiting procedure, involving the splitting-up of the path into "n" segments, where "n" becomes infinite and the lengths of the segments becomes infinitesimal. This reduces to an integral [Transparancy 2] which becomes quite easy to evaluate when the path is parameterized appropriately.
An example was given [Transparancies 3 and 4] in which the value of a path integral is different for two different paths connecting the same two endpoints. This shows that, in general, the value of a path integral between two points depends upon the path taken.
However, there are some types of vector fields, called "conservative" vector fields, in which the path integral depends only upon the endpoints, and is independent of the path used to travel between the two points. These will be considered next lecture.
What is the use of all this? An example was given of work, which is the line integral of a force field.
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)
