Integral theorems for the gradient of a vector field, with a fluid dynamical application

TL;DR

This paper generalizes classical divergence and Kelvin-Stokes theorems through a tensor identity relating volume and surface integrals of a vector field's gradient, with applications in fluid dynamics and discussions on tensor analysis.

Contribution

It introduces a tensor-valued identity that extends classical theorems, providing new insights and applications in fluid dynamics and tensor analysis.

Findings

The identity relates volume integrals of the gradient to surface integrals involving the vector field.

In 2D, it unifies divergence and Kelvin-Stokes theorems with additional strain-related theorems.

Application to oceanic observations demonstrates practical relevance.

Abstract

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.