D/Dt or d/dt for the material derivative? The views of Sir George, Sir Harold, Lady Bertha, Sir James, and other friends.

Known by many names, the material derivative is the total time derivative following the motion of a fluid parcel. The material derivative initially appeared in the works of D'Alembert and Euler in the mid-18th century, when they first formulated hydrodynamics as a field theory governed by partial differential equations (Truesdell, 1954; Darrigol, 2005). At last year's APS-DFD eduation and outreach session, an abstract by Shahbazi (2023) opines that "Using 'D/Dt' to denote the total derivative is awkward and unnecessary as students do not have prior exposure to this notation in the calculus course, and it can be replaced by the familiar notation 'd/dt'." Both notations can be found in the contemporary literature, while some authors elect to write out the full expression every time, without using either of the abbreviations. Introduced by Stokes (1845), the D/Dt notation is criticized by Jeffreys & Jeffreys (1946) as a relic of an obsolete, 19th century notational convention for partial derivatives (see also Cajori, 1929). I present evidence of the painful evolution of material derivative notation from 19th century English, French, and German fluid mechanics textbooks. Meanwhile White (2006) and especially Lighthill (1986) offer defenses of D/Dt that may keep the issue unsettled. An argument can even be made for using d/dt for Lagrangian coordinates, and D/Dt for Eulerian coordinates (Saffman, 1992), echoing Lagrange (1788) and others.