The nabla symbol is the symbol (∇). The name comes from the Arabic meaning arrow, or Hebrew word for a harp, called nevel , which had a similar shape. The symbol was first used by William Rowan Hamilton in the form of a sideways wedge: ⊲.
The nabla is used in mathematics to denote the del operator, a differential operator that indicates taking gradient, divergence, or curl. It also can refer to a connection in differential geometry and to the backward difference operator in the calculus of finite differences, as well as the all relation (most commonly in lattice theory). It was introduced by the Irish mathematician and physicist William Rowan Hamilton in 1837. William Thomson wrote in 1884: "I took the liberty of asking Professor Bell whether he had a name for this symbol and he has mentioned to me nabla, a humorous suggestion of Maxwell's. It is the name of an Egyptian harp, which was of that shape".
In 1901, Josiah Willard Gibbs and Edwin Bidwell Wilson wrote: "This symbolic operator was introduced by Sir W. R. Hamilton and is now in universal employment. There seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated formulae in which occurs a number of times, no inconvenience to the speaker or listener arises from the repetition. V is read simply as 'del V' ".
In abstract interpretation, the nabla is the usual symbol for the widening operator, an operator that permits static analysis of programs to terminate in finite time.
The nabla is used in naval engineering (ship design) to designate the volume displacement of a ship or any other waterborne vessel. Where its counterpart, the Greek delta, is used to designate weight displacement (the total weight of water displaced by the ship), the nabla is used to designate volume displacement, i.e. the total volume of water displaced by the ship.
- Del, the vector differential operator
- Del in cylindrical and spherical coordinates
- grad, div, and curl, differential operators defined using del
- the Covariant derivative, a separate (and tensorial) differential operator defined for tensors
- W. R. Hamilton, in Trans. R. Irish Acad. XVII. 236 (1837)
- W. Thomson, Notes Lect. Molecular Dynamics & Wave Theory of Light at Johns Hopkins Univ. x 112 (MS) (1884)
- Gibbs & Wilson, Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs by Edwin Bidwell Wilson (1901)
- History of Nabla
- A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen