Reference request: A multidimensional generalization of the fundamental theorem of calculus

$$\newcommand\R{\mathbb R}$$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 $$J$$.

Then it is not hard to establish the following multidimensional generalization of the fundamental theorem of calculus (Lemma 5.1): $$$$\int_u^v dx\, f(x)=\sum_{J\subseteq[p]}(-1)^{p-|J|}F(v_J),$$$$ where $$v_J:=\big(v_1\,1(1\in J)+u_1\,1(1\notin J),\dots,v_p\,1(p\in J)+u_p\,1(p\notin J)\big)$$.

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

• Is this not Stokes’ Theorem? – Rivers McForge Jan 12 at 21:54
• @RiversMcForge : I don't see a relation with Stokes' theorem. – Iosif Pinelis Jan 12 at 21:57
• It probably generalizes to polytopes (I know this is not what you are asking, but it is just a comment). – Malkoun Jan 12 at 22:48
• @Iosif Stokes theorem relies the integral of $D_ig$ over, say, the box and the integral of $g$ over it's boundary. This should be Stokes theorem applied $p$ times. – Fedor Petrov Jan 13 at 16:57
• @IosifPinelis I do not know, but I would not expect this: in Stokes theorem the dimensions in LHS and RHS differ by 1, here they differ by $p$. – Fedor Petrov Jan 13 at 21:11

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $$u=(0,0,\ldots,0)$$ and $$v=(1,1,\ldots,1)$$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $$L$$ be a finite set. Let $$f:\mathbb{R}^L\rightarrow \mathbb{R}$$, $$\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$$ be a sufficiently smooth function, and let $$\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$$, then $$f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h}))$$ where $$\psi_A(\mathbf{h})$$ is the element $$\mathbf{x}=(x_{\ell})_{\ell\in L}$$ of $$\mathbb{R}^L$$ defined from the element $$\mathbf{h}=(h_{\ell})_{\ell\in A}$$ in $$[0,1]^A$$ by the rule: $$x_{\ell}=0$$ if $$\ell\notin A$$ and $$x_{\ell}=h_{\ell}$$ if $$\ell\in A$$. Of course one needs to 1) apply this to all $$L$$'s which are subsets of $$[p]$$, 2) use Möbius inversion in the Boolean lattice, and 3) specialize to $$L=[p]$$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $$(\ell_1,\ldots,\ell_k)$$ of arbitrary length of elements of $$L$$, instead of subsets of $$L$$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$$=$$"without repeats", or the stopping rule that one should not tack on an $$\ell$$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $$f$$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

I forgot to mention, one can use Lemma 1 to derive the Taylor formula from calculus 1. This corresponds to $$L$$ having one element and defining allowed sequences as the ones of length at most $$n$$. See

https://math.stackexchange.com/questions/3753212/is-there-any-geometrical-intuition-for-the-factorials-in-taylor-expansions/3753600#3753600

• Thank you for this wealth of information. – Iosif Pinelis Jan 13 at 21:33

The $$p=2$$ dimensional case is an exercise in Rogawski's calculus textbook. It is exercise 47 on page 885, section 15.1 (Integration in Several Variables) in the 2008 Early Transcendentals edition.

• The $p=2$ case is also stated in the proof of Lemma 6.2 of this 1999 paper of Carbery and Wright: mathscinet.ams.org/mathscinet-getitem?mr=1683156 , with the three word proof "By Stokes' theorem". – Terry Tao Jan 13 at 17:10
• Correction: that paper is by Carbery, Christ, and Wright. The general case is implicit in a followup paper mathscinet.ams.org/mathscinet-getitem?mr=1928871 of Katz, Krop, and Maggioni in 2002, proved by "a simple application of the fundamental theorem of calculus". – Terry Tao Jan 13 at 17:30
• @ZachTeitler : Thank you for the reference. – Iosif Pinelis Jan 13 at 20:50
• @TerryTao : Thank you for the references. – Iosif Pinelis Jan 13 at 20:51
• The $p = 2$ case is also well-known for people who work in wave equations (for I hope obvious reasons). For example, it is used lots when studying the spherically symmetric Einstein-scalar-field model in mathematical relativity; I'd say every other paper there uses this formula either explicitly or implicitly. – Willie Wong Jan 14 at 2:17