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

Abstract

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 $$C^1$$ C 1 smoothness of the differential form. As a numerical application, we propose an algorithm for exterior differentiation in $$\mathbb {R}^n$$ 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.