Page 15-2
At any particular point, the maximum rate of change of the function occurs in
the direction of the gradient, i.e., along a unit vector
u
=
∇φ
/|
∇φ
|.
The value of that directional derivative is equal to the magnitude of the
gradient at any point D
max
φ
(x,y,z) =
∇φ
•∇φ
/|
∇φ
| = |
∇φ
|
The equation
φ
(x,y,z) = 0 represents a surface in space. It turns out that the
gradient of the function at any point on this surface is normal to the surface.
Thus, the equation of a plane tangent to the curve at that point can be found
by using a technique presented in Chapter 9.
The simplest way to obtain the gradient is by using function DERIV, available
in the CALC menu, e.g.,
A program to calculate the gradient
The following program, which you can store into variable GRADIENT, uses
function DERIV to calculate the gradient of a scalar function of X,Y,Z.
Calculations for other base variables will not work. If you work frequently in
the (X,Y,Z) system, however, this function will facilitate calculations:
<< X Y Z 3
ARRY DERIV >>
Type the program while in RPN mode. After switching to ALG mode, you can
call the function GRADIENT as in the following example:
Using function HESS to obtain the gradient
The function HESS can be used to obtain the gradient of a function as shown
next. As indicated in Chapter 14, function HESS takes as input a function of
