Once you define functionals,
and then define grad of 'em,
getting vector fields,

the natural question is:
"Can you go backwards?"
That is, is a given vector field grad of some function?
(If so, call it a "potential function")

Obstructions:
- local
  (has to be locally a derivative,
   i.e., curl-free)
- global
  this is the topology!
  can't climb a spiral staircase!
...and this exactly yields homology:

(possible)/(achievable) = obstruction
i.e.,
(local obs. vanishes) / (globally possible) = global obstruction
----------------------------------------------------
This is a -fundamental- technical tool in physics:
it replaces thinking about -forces- with thinking about -energy-
(eg, this is how Feynman does mechanics).
This is -not- how Newton thought of it -- it's a refinement.
[Just as Adam Smith didn't think of supply and demand curves]

EG of computing change in elevation (in "24 differential forms.tex")

EG of electrical potential vs. electrical force

This also leads into PDEs and fundamental solutions
(a great way to get lots of integrable fields).
