Begin2.DVI

of stationary values of f with a constraint condition because one would have at a

stationary value of F the conditions

∂F

∂x

=∂f

∂x

+λ∂g

∂x

=0

∂F

∂y

=∂f

∂y

+λ∂g

∂y

=0

∂F

∂λ

=g(x, y ) =0 The constraint condition.

(7 .81)

These represent three equations in the three unknowns x, y, λ that must be solved.

The equations (7.80) and (7.81) are known as the Lagrange rule for the method of

Lagrange multipliers.

The method of Lagrange multipliers can be

applied in higher dimensions. For example,

consider the problem of finding maximum

and minimum values associated with a func-

tion f =f(x, y, z ) subject to the constraint

conditions g(x, y, z ) = 0 and h(x, y, z) = 0 .Here

the equations g(x, y, z ) = 0 and h(x, y, z) = 0

describe two surfaces that may or may not

intersect. Assume the surfaces intersect to

give a space curve.

The problem is to find an extremal value of f =f(x, y, z )as (x, y, z )varies along

the curve of intersection of surfaces g = 0 and h= 0. At a critical point where a

stationary value exists, the directional derivative of falong this curve must be zero.

Here the directional derivative is given by df

ds

=∇f·ˆet,where ˆetis a unit tangent

vector to the space curve and ∇f=grad f denotes the gradient of f. Note that if

the directional derivative is zero, then ∇f must lie in a plane normal to the curve of

intersection.

Another way to view the problem, and also suggest that the concepts can be

extended to higher dimensional spaces, is to introduce the notation ̄x= (x 1 , x 2 , x 3 ) =

(x, y, z )to denote a vector to a point on the curve of intersection of the two surfaces

g(x 1 , x 2 , x 3 ) = 0 and h(x 1 , x 2 , x 3 ) = 0. At a stationary value of f one must have

df = ∂f

∂x 1

dx 1 + ∂f

∂x 2

dx 2 + ∂f

∂x 3

dx 3 =grad f·dx ̄= 0