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.
Fields of Study & Subjects
Mathematical and Theoretical Analysis
Mathematics and Applications
Advanced Banach Space Theory
Publication details
- Journal / source
- São Paulo Journal of Mathematical Sciences
- Publisher
- Springer Science and Business Media LLC
- Volume
- 20
- Issue
- 2
- Publication date
- 18 September 2026
- Language
- EN