  A closed expression of the Euclidean Wilson-loop functionals is derived for
pure Yang-Mills continuum theories with gauge groups $SU(N)$ and $U(1)$ and
space-time topologies $\Rl^1\times\Rl^1$ and $\Rl^1\times S^1$. (For the $U(1)$
theory, we also consider the $S^1\times S^1$ topology.) The treatment is
rigorous, manifestly gauge invariant, manifestly invariant under area
preserving diffeomorphisms and handles all (piecewise analytic) loops in one
stroke. Equivalence between the resulting Euclidean theory and and the
Hamiltonian framework is then established. Finally, an extension of the
Osterwalder-Schrader axioms for gauge theories is proposed. These axioms are
satisfied for the present model.
