Let $U$ be an open subset of $N_{\mathbb{R}}$ and let $\sigma$ be an $r$-dimensional integral ${\mathbb{R}}$-affine polyhedron contained in $U$. For any closed face $\rho$ of codimension $1$, let $\omega_{\rho,\sigma}:=\omega_{\partial H,H}$ using 2.7 for the affine hyperplane $\partial H$ generated by $\rho$ and the corresponding halfspace containing $\sigma$. We note that $\omega_{\rho,\sigma}\in N$ is determined up to addition with elements in $N_{\rho}=N\cap{\mathbb{L}}_{\rho}$, where ${\mathbb{L}}_{\rho}$ is the linear hyperplane parallel to $\rho$.

For $\eta\in A_{c}^{r-1,r}(U)$, we have introduced the contraction $\langle\eta;\omega_{\rho,\sigma}\rangle_{\{r\}}$ as an element of $A_{c}^{r-1,r-1}(U)$ which is obtained by inserting the vector $\omega_{\rho,\sigma}$ for the $r$-th argument of the corresponding multilinear function (see 2.6). Note that the restriction of this contraction to $\rho$ does not depend on the choice of the representative $\omega_{\rho,\sigma}$. Then we define

$$\int_{\partial\sigma}\eta:=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma}\rangle_{\{r\}},$$

where $\rho$ ranges over all closed faces of $\sigma$ of codimension $1$. On the right, we use the integrals of $(r-1,r-1)$-superforms from 2.4. For $\eta\in A_{c}^{r,r-1}(U)$, we define similarly

$$\int_{\partial\sigma}\eta:=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma}\rangle_{\{1\}}.$$

Note that the integrals do depend only on the integral ${\mathbb{R}}$-affine structure of $N_{\mathbb{R}}$ but do not depend on the choice of the orientation of $N_{\mathbb{R}}$.

If $\sigma$ is an integral ${\mathbb{R}}$-affine polyhedron of any dimension $n$ and if $\eta\in A_{c}^{n-1,n}(U)$ for an open subset $U$ of $N_{\mathbb{R}}$ containing $\sigma$, then we define $\int_{\partial\sigma}\eta$ by applying the above to the affine space ${\mathbb{A}}_{\sigma}$ generated by $\sigma$ and to the pull-back of $\eta$ to ${\mathbb{A}}_{\sigma}$. We give now a concrete description of $\int_{\partial\sigma}\eta$ in terms of integrals over classical $n-1$-forms. For every closed face $\rho$ of $\sigma$, let $N_{\sigma}={\mathbb{L}}_{\sigma}\cap N$ be the canonical integral structure on the affine space generated by $\sigma$. If $e_{1}^{\rho},\dots,e_{n-1}^{\rho}$ is a basis of $N_{\rho}$, then $\omega_{\rho,\sigma},e_{1}^{\rho},\dots,e_{n-1}^{\rho}$ is a basis of $N_{\sigma}$. We note that the contraction $\langle\eta;\omega_{\rho,\sigma},e_{1}^{\rho},\dots,e_{n-1}^{\rho}\rangle_{\{n,\dots,2n-1\}}$ may be viewed as a classical $(n-1)$-form on $U$ and hence we get

$$\int_{\partial\sigma}\eta=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma}\rangle_{\{n\}}=\sum_{\rho}\int_{\rho}\langle\eta;\omega_{\rho,\sigma},e_{1}^{\rho},\dots,e_{n-1}^{\rho}\rangle_{\{n,\dots,2n-1\}}.$$