An integral theorem for the velocity gradient

A nearly forgotten 140 year old mathematical result can help extract new value from oceanographic observations, as we show in a new paper, Lilly, Feske, Fox-Kemper, and Early (2024), recently accepted to Proceedings of the Royal Society of London, Series A. Integral theorems linking the spatial derivatives of a vector field to its values along a boundary play a central role in physics. Yet a generalized theorem, dating back to J. W. Gibbs and the origins of vector calculus in the late 1800’s, is little known today. We re-examined this gradient tensor theorem, its interpretation, and its expression in modern notational systems. Using this result, observations of an oceanic vortex by a single moving platform, such a ship, that traces out closed cells (the triangles) can be used to infer spatially-averaged rates of strain—important measures of small-scale deformation that are otherwise challenging to observe.