Completed discussion on potentials in 3 dimensions.
Previously we found that a force is conservative, allowing the
potential to be unambiguously defined, only if the work integral
between any two points is path-independent. We have shown this occurs
only if the work integral around any closed path vanishes. Last
time we showed -- by deriving Stokes theorem -- that all closed work
integrals vanish if and only if 
vanishes.
We reviewed the physical meaning of the gradient 
, an operator
acting on a multivariable function 
to give the vector
which points in the direction of maximal change of f, and
perpendicular to any contours on which f is constant. We noted that
the gradient contains all information about first order changes in the
function, as we vary the argument 
in any direction. Thus
(as you will see in homework) directional derivatives of f in any
direction 
are given by the dot product of
with the gradient .
We then discussed calculating the gradient in other coordinate
systems, by insisting on the geometric definition 
. 
can be obtained computationally (as the velocity
times dt), or geometrically, as the physical displacement induced by
infinitesimal changes in coordinates. This geometric construction of
df using the gradient must agree with the chain rule for the
differential of f, a multivariable function of our coordinates. Thus
the gradient operator is easily constructed in various coordinate
systems.
Finally, we showed that our work integral definition of the potential
implied that 
. This led to our final,
equivalent, way of stating that a force is conservative: a force is
conservative if and only if it can be written 
, for some function g of position.
-- KB