GRADIENT, DEL OPERATOR, DIRECTIONAL DERIVATIVE

Def. Gradient. Let Φ(x, y, z) be a scalar point function that is defined and differentiable at each point (x, y, z) in a certain region of space. The gradient of Φ is a vector point function defined as

The gradient of Φ is also denoted by

where is called the del operator.

Stated in words, the gradient of a scalar point function Φ(x, y, z) is a vector whose components along the x, y, z axes are the partial derivatives of Φ(x, y, z) with respect to the variables.

Def. Del operator. The del operator is defined by

something that has no meaning by itself. It is a contrivance that is useful in working with vector functions. It is used in defining the gradient, divergence and curl and is a thing which is convenient because it can be treated as if it were a vector in working with vector functions. It obeys vector laws and when combined with vectors gives meaningful results. It is a convenient device. It is also known as nabla.

Gradient. Physical interpretation. The gradient, at any point P:(x, y, z), of a scalar point function Φ(x, y, z) is a vector that is normal to that level surface of Φ(x, y, z) that passes through point P. The magnitude of the gradient is equal to the rate of change of Φ (with respect to distance) in the direction of the normal to the level surface at point P.

Grad Φ, evaluated at a point P:(x₀, y₀, z₀), is normal to the level surface Φ(x, y, z) = c passing through point P. The constant c is given by c = Φ(x₀, y₀, z₀).

Directional derivative of a scalar point function. The rate of change (i.e. derivative) of a scalar point function Φ in some specified direction is called the directional derivative in that direction.

The rate of change (with respect to distance) of Φ(x, y, z) at a point P in some specified direction is as follows: Let the direction be specified by a unit direction vector a. Then the rate of change of Φ in the direction of a is given by

                                    grad Φ∙ a

i.e. the dot product of grad Φ and a. In other words,

where

is the directional derivative of Φ in the direction of unit vector a.

Proof

Thus the rate of change of Φ in the direction of a unit vector a is the component of grad Φ in the direction of a (i.e. the projection of grad Φ onto a ). See Fig. 1. The maximum value of the directional derivative occurs when the directional vector a coincides with the direction of grad Φ. Thus the directional derivative achieves its maximum in the direction of the normal to the level surface Φ(x, y, z) = c at P.