Noticing that,

Note that the potential energy function U depends on the reference xyz used or the datum used. However, the change in potential energy is independent of the datum used.

Change in potential energy , of a force field is the negative of the work done by the force field on a particle in going from position 1 to position 2 along any path. For any closed path , clearly the work done by a conservative force field F is

For an infinitesimal path difference dr starting from 1,

F.dr = -dU

F_xdx+F_ydy+F_zdz