The del operator, denoted with what is called the nabla symbol (an inverted delta), is a differential operator connecting differential calculus of functions to the study of vectors, and vector-valued functions. The del operator has several forms, and is defined by
where ∂/∂x indicates the partial derivative with respect to x (similarly for y and z), and i, j, and k indicate the three standard unit vectors (all in the Cartesian coordinate system).
To a scalar function, F(x,y,z), in the Cartesian coordinate system, the del operator may be applied to create what is known as the gradient of F — defined as
The inclusion of the unit vectors in ∇ lead to the gradient’s being a vector-valued function in three dimensions, and the vector consequently is directed toward the greatest increase in F, at any point (x,y,z). Its magnitude is equal to the maximum rate of increase, and hence may be used as an analog to the 1-dimensional derivative, in 3 dimensions.
When ∇ is applied to a scalar field (function) of the form F(x,y,z), and a vector field a (= <a_1, a_2, a_3>) is chosen, the directional derivative takes the form
in which the dot-product of the gradient of F and and the vector a is taken. The most common analogy is this: if ∂F/∂x gives the rate of change of F in the x direction, then ∇F • a gives the rate of change of F, in vector form, toward the vector a. a is taken to be the unit vector for this calculation. This operation allows the rate of change of a scalar field with respect to an arbitrary — and sometimes changing — direction of a vector (a need not be a vector composed only of constant components), to be calculated. Its most common application for this operation lies in the field of fluid dynamics.
Coming soon: ∇applied to vector functions!