Integral of a Differential Form over an Oriented k-Sub-Rectangle in Euclidean Space

Let $n,k\in\mathbb{N}$ with $k\le n$, let $(S,\varepsilon)$ be an oriented $k$-sub-rectangle of $\mathbb{R}^n$, let $U\subseteq \mathbb{R}^n$ be open with $S\subseteq U$, and let $\omega$ be a continuous differential $k$-form on $U$. Choose data as in Oriented k-Sub-Rectangle of Euclidean Space, so that $S=\lambda_R(R)$ for a standard $k$-rectangle

and active coordinate indices $1\le i_1<\cdots<i_k\le n$. Write $\omega$ in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. Let $f:U\to\mathbb{R}$ be the coefficient function of the basis form

For each fixed $(t_2,\dots,t_k)\in [a_2,b_2]\times\cdots\times[a_k,b_k]$, define

where this is the one-dimensional Riemann integral on $[a_1,b_1]$; it exists because the integrand is continuous on $[a_1,b_1]$, hence Riemann integrable. Recursively, for each $r\in\{2,\dots,k\}$ and each fixed $(t_{r+1},\dots,t_k)\in [a_{r+1},b_{r+1}]\times\cdots\times[a_k,b_k]$, define

again in the one-dimensional Riemann sense of Riemann Integrability on a Closed Interval, whenever the integrand is viewed as a function of $t_r$ alone. The integral of $\omega$ over $(S,\varepsilon)$ is defined by

Equivalently,

with the right-hand side understood through the recursive construction above.