The exterior derivative and the mean value equality in $\mathbb{R}^n$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911539606323200 |
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| author | Fadel, Daniel Earp, Henrique N. Sá Silva, Tomás S. R. |
| author_facet | Fadel, Daniel Earp, Henrique N. Sá Silva, Tomás S. R. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The exterior derivative and the mean value equality in $\mathbb{R}^n$ Fadel, Daniel Earp, Henrique N. Sá Silva, Tomás S. R. Differential Geometry Numerical Analysis 26B05, 58A10, 65D25 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. |
| title | The exterior derivative and the mean value equality in $\mathbb{R}^n$ |
| topic | Differential Geometry Numerical Analysis 26B05, 58A10, 65D25 |
| url | https://arxiv.org/abs/2510.00999 |