The exterior derivative and the mean value equality in $\mathbb{R}^n$

TL;DR

The paper addresses extending the Mean Value Theorem and Stokes' theorem to differential forms in $\mathbb{R}^n$ by reinterpreting the exterior derivative as an infinitesimal flux $D$. It develops a generalized Trisection Lemma for $k$-blocks, introduces flux-continuity and flux-integrability, and proves a mean-value equality $D\omega_\xi(e_1,\dots,e_k) = \frac{1}{\mathrm{vol}(B)} \int_{\partial B}\omega$, enabling a Stokes theorem under weaker regularity. Together with a consistency result $D=\mathrm{d}$ in the differentiable case, these ideas yield both a conceptual unification of FTC, MVT, and Stokes in higher dimensions and a practical mesh-free numerical method for exterior differentiation. The results offer both conceptual insights into multivariable calculus and practical tools for computation on flux-based differentiation.

Abstract

This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional analogue of the Mean Value Theorem, valid for differential $k$-forms, and provides a natural formulation of Stokes' theorem that mirrors the exact hypotheses of the Fundamental Theorem of Calculus -- without requiring full $C^1$ smoothness of the differential form. As a numerical application, we propose an algorithm for exterior differentiation in $\mathbb{R}^n$ that relies solely on black-box access to the differential form, offering a practical tool for computation without the need for mesh discretization or explicit symbolic expressions.

Theorems & Definitions (39)

• Lemma 1: Trisection Lemma
• Lemma 2
• Theorem 1: MVT for differential forms
• Theorem 2: Stokes without $C^1$ assumption
• proof : Proof of Lemma \ref{['lemma: fundamental']}
• Theorem 3: M.V.T., $1$-dimensional case
• Remark 1
• proof
• Theorem 4: Second Fundamental Theorem of Calculus
• proof