  A general action is proposed for the fields of $q$-dimensional differential
form over the compact Riemannian manifold of arbitrary dimensions. Mathematical
tools are based on the well-known de Rham-Kodaira decomposing theorem on
harmonic integral. A field-theoretic action for strings, $p$-branes and
high-spin fields is naturally derived. We also have, naturally, the generalized
Maxwell equations with an electromagnetic and monopole current on a curved
space-time. A new type of gauge transformations ({\it dual} gauge
transformations) plays an essential role for coboundary $q$-forms.
