(31)

Now the constraint g(x)=0 defines an n-1-dimensional surface in En. The unit vector that is normal to that surface is the vector

where the gradient of g(x) is the vector

and where the vectors ei, i=1, ···, n, are an orthonormal basis of E n. Since

we can write the directional derivative in the more compact manner

Consider some vector u that is tangent to the surface defined by g(x)=0. Since n is the unit normal to that surface, it is necessary that

But consider f(x). If x is a point on the surface, then, to find an extremum of f(x), we must move in a direction tangent to the surface. That is, when f(x) is at an extremum, the directional derivative in any direction perpendicular to n must be zero. Therefore, for any vector u perpendicular to n,

We can write each basis vector ei as the sum of a vector parallel to n and a vector

perpendicular to n. The parallel component is (ei·n)n; the perpendicular component is

since n is a unit vector, n·n=1, and the reader can verify that wi·n=0. The derivative of f(x) in the direction of wi is

Since

and

we can write

for i=1, …, N. The parameter ? is given by

This shows that solving the system of equations (4) is equivalent to finding the extrema of f(x) subject to the constraint g(x)=0.