From the Rank Theorem to the Vaschy-Buckingham Theorem
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901239516626944 |
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| author | Olivier Ky Thiêp CHOFFRUT-PHAN |
| author_facet | Olivier Ky Thiêp CHOFFRUT-PHAN |
| contents | <pre>This pedagogical note presents the mathematical foundations of the Vaschy-Buckingham theorem, known in fluid mechanics as the $\Pi$-theorem. The goal is to make explicit the link between this central result of dimensional analysis and the rank theorem of a linear map, as taught in first- or second-year undergraduate linear algebra. We show that the dimensionless condition can be reformulated as the search for the kernel of a linear map --- the dimensional matrix --- and that the number of independent dimensionless quantities is precisely the dimension of this kernel. The running example is the Stokes problem: the viscous drag exerted by a Newtonian fluid on a moving sphere, for which the method recovers the canonical form $F = \varphi(Re)\,\rho v^2 R^2$, bringing out the Reynolds number.</pre> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19428776 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | From the Rank Theorem to the Vaschy-Buckingham Theorem Olivier Ky Thiêp CHOFFRUT-PHAN linear algebra rank theorem kernel dimensional analysis Vaschy-Buckingham theorem Pi theorem dimensionless numbers Stokes problem physics education mathematical physics <pre>This pedagogical note presents the mathematical foundations of the Vaschy-Buckingham theorem, known in fluid mechanics as the $\Pi$-theorem. The goal is to make explicit the link between this central result of dimensional analysis and the rank theorem of a linear map, as taught in first- or second-year undergraduate linear algebra. We show that the dimensionless condition can be reformulated as the search for the kernel of a linear map --- the dimensional matrix --- and that the number of independent dimensionless quantities is precisely the dimension of this kernel. The running example is the Stokes problem: the viscous drag exerted by a Newtonian fluid on a moving sphere, for which the method recovers the canonical form $F = \varphi(Re)\,\rho v^2 R^2$, bringing out the Reynolds number.</pre> |
| title | From the Rank Theorem to the Vaschy-Buckingham Theorem |
| topic | linear algebra rank theorem kernel dimensional analysis Vaschy-Buckingham theorem Pi theorem dimensionless numbers Stokes problem physics education mathematical physics |
| url | https://doi.org/10.5281/zenodo.19428776 |