Mathematics of Physics and Engineering

Functions, Sets, Gradient 129

we conclude that the vector Ax + 12 j — k defines the plane tangent to the
surface at the point PQ = (1,2,14). The equation of the plane at PQ is
4x + 12y — z = 14 : note that x = 1, y = 2, z = 14 must satisfy the
equation, and you get the number 14 on the right by plugging in these
values for x,y,z.

EXERCISE 3.1.11. B Verify that r(u,v) = ui + vj + (2u^2 + 3v^2 )k and
r(u,v) = (v/y/2) cosuz4- (v/y/3) sinuj+v^2 k are two possible parametric
representations of the surface z = 2a:^2 4- 3y^2. Verify that, for both repre-
sentations, the vector ru x rv at the point (1,2,14) is parallel to the vector
4i+12j- k.

Recall that P is a critical point of a scalar field / if either V/(Po) —

0 or / is defined but not differentiable at PQ. HP is not a critical point of

/, then — V/(P) is the direction of the most rapid decrease of the function

/ at the point P. This observation suggests the following iterative method

for finding the point of local minimum of /: given a reference point O, a

starting point Pi, and a collection of positive numbers ak, k = 2,3,...,

define the points Pj, recursively by

OPk+i =OPk- ofcV/(Pfc). (3.1.12)

Sometimes it is convenient to identify the point Pfc with the position vector

rfc = OPk and to write (3.1.12) in an equivalent form as

T-fc+i =rfc -akVf(rk)-

The new point Pk+\ can be denned as long as V'/(Pfc) ^ 0. If the sequence

of the points Pk, k > 1, converges as k —> oo, then the limit P* must be the

point of local minimum of /; otherwise V/(P*) ^ 0 and the sequence can

be continued. As a result, V/(Pfc) —> 0, and in practice, the computations

are stopped as soon as ||V/(Pfc)|| is sufficiently small. Special choices of the

numbers ak can speed up the convergence. This method is known as the

method of steepest descent. It was one of the first numerical methods

for minimizing a function. The method can converge very slowly, but is

easy to implement. There exist more sophisticated numerical methods for

finding critical points.

EXERCISE 3.1.12. A (a) Draw a picture illustrating (3.1.12). (b) Write

a computer program that implements (3.1.12) to find the local minimum

of the function f(x,y) = 2x^2 + 3y^2. Experiment with various starting

points, various values of ak and different stopping criteria (for example,