  For an integer $m\geq 1$, a combinatorial manifold $\widetilde{M}$ is defined
to be a geometrical object $\widetilde{M}$ such that for $\forall
p\in\widetilde{M}$, there is a local chart $(U_p,\phi_p)$ enable $\phi_p:U_p\to
B^{n_{i_1}}\bigcup B^{n_{i_2}}\bigcup...\bigcup B^{n_{i_{s(p)}}}$ with
$B^{n_{i_1}}\bigcap B^{n_{i_2}}\bigcap...\bigcap
B^{n_{i_{s(p)}}}\not=\emptyset$, where $B^{n_{i_j}}$ is an $n_{i_j}$-ball for
integers $1\leq j\leq s(p)\leq m$. Integral theory on these smoothly
combinatorial manifolds are introduced. Some classical results, such as those
of {\it Stokes'} theorem and {\it Gauss'} theorem are generalized to smoothly
combinatorial manifolds in this paper.
