A fundamental piece of dogma that permeates much of current science, is
expressed by the statement:
"A scientific theory (of merit) must be able to predict a unique outcome,
given initial data." However, consider smooth evolutionary maps from an initial to a final state and impose the condition that the Jacobian is not invertible, such that the evolutionary process is not a homeomorphism (and therefore the topology of the final state is not the same as the topology of the initial state.). Then, in contradiction to the dogma above, given initial state vector field functions, it is IMPOSSIBLE to deterministically PREDICT final state vector field functions. Note that it is the functional form in the neighborhood of a point, not the value of the field at the point, that is the I>unpredictable concept.
However, the situation is changed remarkably when the alternate question is
asked. Given covariant vector field functions, or contravariant vector density
functions, on the final state, is it possible to deterministically retrodict the
vector field functions on the initial state, even though topology is not
preserved during the evolutionary process, and the map is irreversible? The
anwser is YES, retrodiction of the functional form of covariant tensor fields is
possible, but prediction of tensor field function form is not possible, if the
map is not a homeomorphism.
The fact that Pair (with covariant coefficients) and Ampair ( with contravaraint
densities as coefficients) exterior differential forms are well behaved with
respect to pullbacks for C1 smooth nonhomeomorphic evolutionary maps, is
perhaps the most fundamental reason for studying
HREF="difforms.html">Cartan's exterior calculus.
