The Operator ∇ and Its Uses

The differential operator ∇ was introduced by Sir William Rowan Hamilton and developed by P. G. Tait; it is of central importance in many three-dimensional physical problems. The symbol ∇ was originally named ‘nabla’ after a harp-like ancient Assyrian musical instrument of similar shape; it is now usual to adopt the term ‘del’ introduced by J. Willard Gibbs. In cartesian notation 4.1 $$ del = \nabla = i\frac{\partial }{{\partial x}} + j\frac{\partial }{{\partial y}} + k\frac{\partial }{{\partial z'}} $$ and may be applied as an operator either to a scalar or to a vector function of space. Again, treating the differentiators in ∇ as scalare, we may formally regard ∇ as a vector which can have either a scalar or a vector product with other vectors. In vector analysis there are three fundamental operations with ∇ which are of physical interest. If S is a scalar function and V a vector function of position, these operations are (i) ∇S, where ∇ acts as an operator; (ii) ∇. V, and (iii) ∇ × V, where ∇ is treated formally as a vector.