$\newcommand\R{\mathbb R}$Let $f\colon\R^p\to\R$ be a continuous function. For $u=(u_1,\dots,u_p)$ and $v=(v_1,\dots,v_p)$ in $\R^p$, let $[u,v]:=\prod_{r=1}^p[u_r,v_r]$; $u\wedge v:=\big(\min(u_1,v_1),\dots,\min(u_p,v_p)\big)$; $u\vee v:=\big(\max(u_1,v_1),\dots,\max(u_p,v_p)\big)$; $$\int_u^v dx\, f(x):= (-1)^{\sum_{r=1}^p\,1(u_r>v_r) }\int_{[u\wedge v,u\vee v]}dx\,f(x).$$ Let $F\colon\R^p\to\R$ be any antiderivative of $f$, in the sense that $$D_1\cdots D_p F=f,$$ where $D_j$ is the operator of the partial differentiation with respect to the $j$th argument; it is assumed that the result of this repeated partial differentiation does not depend on the order of the arguments with respect to which the partial derivatives are taken. Let $[p]:=\{1,\dots,p\}$. For each set $J\subseteq[p]$, let $|J|$ denote the cardinality of $J$, and also let
$1_J:=\big(1(1\in J),\dots,1(p\in J)\big)$.

Then it is not hard to establish the following multidimensional generalization of the fundamental theorem of calculus: \begin{equation} \int_u^v dx\, f(x)=\sum_{J\subseteq[p]}(-1)^{p-|J|}F(v_J), \end{equation} where $v_J:=u+(v-u)\,1_J$.

Has anyone seen this or similar statement elsewhere? (I am only asking about references, not proofs.)