Integral theorems for the gradient of a vector field, with a fluid dynamical application
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913216534151168 |
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| author | Lilly, Jonathan M. Feske, Joel Fox-Kemper, Baylor Early, Jeffrey |
| author_facet | Lilly, Jonathan M. Feske, Joel Fox-Kemper, Baylor Early, Jeffrey |
| contents | The familiar divergence and Kelvin-Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the outer product of the vector field with the exterior normal. The importance of this long-established yet little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the 2D divergence and Kelvin-Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential generalization of the generalized identity to curved two-dimensional surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_13157 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integral theorems for the gradient of a vector field, with a fluid dynamical application Lilly, Jonathan M. Feske, Joel Fox-Kemper, Baylor Early, Jeffrey Fluid Dynamics Mathematical Physics The familiar divergence and Kelvin-Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the outer product of the vector field with the exterior normal. The importance of this long-established yet little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the 2D divergence and Kelvin-Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential generalization of the generalized identity to curved two-dimensional surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems. |
| title | Integral theorems for the gradient of a vector field, with a fluid dynamical application |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2309.13157 |