Having pinned down the meaning of the right hand side of Newton II
(acceleration), we returned to the full problem of solving Newton II
in 3 space dimensions. We noted that for forces which are constant or
dependent only on time, the vector equation just gives 3 differential
equations, 
, where i labels the component along
the i-th basis vector. Note that these equations can be coupled, as
the component of acceleration in the i-th direction is not
always merely the second time derivative of 
(the position
component in the i-th direction), as we showed for polar
coordinates.
We then generalized the work-energy theorem to three dimensions,
obtaining for the power the dot product between the force applied and
the velocity of the particle. For the work, we obtained a path
integral of the force applied along the direction of the particle's
motion. We talked about extending our 1-dimensional concept of
potential energy to 3 dimensions, by defining potential energy at
as the work done to bring the particle from a standard
location 
to . Note, however, that our definition
of the work performed involves the path followed by the particle as we
move it between the two endpoints. We might reasonably expect the
result obtained to depend on which path we chose in bringing the
particle from to . To define a meaningful
potential, which depends only on and not on a complete history of
the particle's motion, we insist that our answer not depend on the
path taken. We showed that this condition (path independence of the
line integral for the potential) was equivalent to requiring that the
line integral for work done around any closed loop vanish.
To get some practice and to derive Stokes' theorem for path integrals,
we began to calculate explicitly the work performed by a force field
(in Cartesian coordinates) on a particle that
follows the following rectangular path in the xy-plane: 
.
By working carefully, always taking the component of force along the
integration variable which locally describes the path, we were able to
relate the path integral to a sum of ordinary integrals, one for each
leg of the rectangular path. We will relate our sum to area integrals
next time.
-- KB