The Homological Unification of Fundamental Theorems of Calculus
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2025
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| _version_ | 1866901577913073664 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the deep conceptual and structural unity underlying the various fundamental theorems of calculus through the lens of modern algebraic topology, specifically homological algebra and differential forms. Starting from the classical Fundamental Theorem of Calculus (FTC), we extend the discussion to Green's Theorem, Stokes' Theorem, and Gauss's Divergence Theorem, demonstrating how each is a specific instance of a single overarching principle: the generalized Stokes' Theorem. We introduce the necessary machinery of differential forms, exterior calculus, and the concept of a boundary operator on manifolds, revealing how the act of differentiation on a form and integration over a manifold boundary are inextricably linked. This homological perspective provides a powerful framework for understanding not only the intrinsic beauty of these theorems but also their profound implications across mathematics, physics, and engineering. The paper elucidates how the abstract language of cohomology groups provides a natural setting for these theorems, offering a coherent and elegant unification that transcends their seemingly disparate classical formulations. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17690592 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Homological Unification of Fundamental Theorems of Calculus SÉRGIO DE ANDRADE, PAULO This paper explores the deep conceptual and structural unity underlying the various fundamental theorems of calculus through the lens of modern algebraic topology, specifically homological algebra and differential forms. Starting from the classical Fundamental Theorem of Calculus (FTC), we extend the discussion to Green's Theorem, Stokes' Theorem, and Gauss's Divergence Theorem, demonstrating how each is a specific instance of a single overarching principle: the generalized Stokes' Theorem. We introduce the necessary machinery of differential forms, exterior calculus, and the concept of a boundary operator on manifolds, revealing how the act of differentiation on a form and integration over a manifold boundary are inextricably linked. This homological perspective provides a powerful framework for understanding not only the intrinsic beauty of these theorems but also their profound implications across mathematics, physics, and engineering. The paper elucidates how the abstract language of cohomology groups provides a natural setting for these theorems, offering a coherent and elegant unification that transcends their seemingly disparate classical formulations. |
| title | The Homological Unification of Fundamental Theorems of Calculus |
| url | https://doi.org/10.5281/zenodo.17690592 |