Continued discussion on potentials and path integrals in 3 dimensions.
Last time we found that the potential could be defined only if the
work integral between any two points was path-independent -- which
could be true only if the work integral around any closed path
vanished. (Such a closed path could be obtained as 
, where
and 
are two different paths between the two endpoints;
requiring the work integrals along and to be equal forces the
integral along the closed path to vanish.)
Today we continued our calculation of the work performed by a force
field 
on a particle that follows the path in the
xy-plane: 
. By working carefully, always taking the
component of force along the integration variable which locally
describes the path, and then applying the fundamental theorem of
calculus, we showed the following. We equated the work integral around
the path to an area integral, over the enclosed rectangle, of the
integrand 
. We then displayed Stokes' theorem, of which our calculation was a
special case. Stokes' theorem states that the work integral around any
closed curve, lying in an arbitrary plane perpendicular to some unit
vector 
, is equivalent to the area integral, over the enclosed
surface, of an integrand given by the dot product of with the curl
of F. We defined the curl in terms of the gradient operator, which we
defined in Cartesian coordinates as 
.
We came away from Stokes' theorem with two facts: 1) we derived a relationship between a closed path integral and an integral over the enclosed area. A similar analysis works in one higher dimension to relate surface flux integrals to integrals over the enclosed volume; this is the divergence theorem (or Gauss' law relating electric flux to charge density integrals). 2) It tells us that our condition for a force to be conservative (so that the potential can be defined unambiguously), which we stated as " the work integral around any closed path must vanish", could equivalently be stated "the curl of F must vanish". Discussed how a non-vanishing curl, or work integral, would lead to energy nonconservation by pumping work into particles that repeatedly circle a closed path.
We then discussed the physical meaning of the gradient, as an operator
acting on a function of position f(r) to give a vector 
. This
vector points in the direction of maximal change of f, and is
perpendicular to any contours on which f is constant.
-- KB