What is path independence?



Today in class we went over the implications set forth by the Fundamental Theorem of Line Integrals. For conservative functions, this theorem actual makes otherwise very long problems very easy by simply finding the potential function and then subtracting the function of the beginning point from the function of the end point. We observed that the following are equivalent (refered as T.F.A.E.)

F is conservative <=> F dot dr is path independent <=> Integral over c of F dot dr = 0 for every closed curve

This gives rise to a new integral: a closed-curve (or closed-loop) integral, which can be seen to the left.

We also tested path independence, which is proved by the Fundamental Theorem of Line Integrals: no matter what path you take, the amount of work done is soley based on the beginning and end points (as long as F is conservative)