  I construct a global version of the local polysymplectic approach to
covariant Hamiltonian field theory pioneered by C. Gunther. Beginning with the
geometric framework of the theory, I specialize to vertical vector fields to
construct the (poly)symplectic structures, derive Hamilton's field equations,
and construct a more-or-less natural Poisson bracket. I then examine a few key
examples to determine the nature of the necessary vertical projections and find
that the theory provides the geometric analog of the canonical transformation
approach to covariant Hamiltonian field theory advanced by Struckmeier and
Redelbach. I conclude with a few remarks about possible applications of this
framework to the geometric quantization of classical field theories.
