  Let $F: [0, \infty) \to [0, \infty)$ be a strictly increasing $C^2$ function
with $F(0)=0$. We unify the concepts of $F$-harmonic maps, minimal
hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and
introduce $F$-Yang-Mills fields, $F$-degree, $F$-lower degree, and generalized
Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on
manifolds. When $F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,$ and
$1-\sqrt{1-2t},$ the $F$-Yang-Mills field becomes an ordinary Yang-Mills field,
$p$-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus
sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a
manifold respectively. We also introduce the $E_{F,g}-$energy functional (resp.
$F$-Yang-Mills functional) and derive the first variational formula of the
$E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) with
applications. In a more general frame, we use a unified method to study the
stress-energy tensors that arise from calculating the rate of change of various
functionals when the metric of the domain or base manifold is changed. These
stress-energy tensors, linked to $F$-conservation laws yield monotonicity
formulae. A "macroscopic" version of these monotonicity inequalities enables us
to derive some Liouville type results and vanishing theorems for $p-$forms with
values in vector bundles, and to investigate constant Dirichlet boundary value
problems for 1-forms. In particular, we obtain Liouville theorems for
$F-$harmonic maps (e.g. $p$-harmonic maps), and $F-$Yang-Mills fields (e.g.
generalized Yang-Mills-Born-Infeld fields on manifolds). We also obtain
generalized Chern type results for constant mean curvature type equations for
$p-$forms on $\Bbb{R}^m$ and on manifolds $M$ with the global doubling property
by a different approach. The case $p=0$ and $M=\mathbb{R}^m$ is due to Chern.
