| Σ | δu | δx1 + λ1 Σ | δu1 | δx1 + ... + λm Σ | δum | δx1. |
| δx1 | δx1 | δx1 |
We can choose λ1, ... λm, to make the coefficients of δx1, δx2, ... δxm, vanish, and the remaining δxm+1 to δxn may be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put
| δu | + | δu1 | + ... + λm | δum | = 0 |
| δxr | δxr | δxr |
for all values of r. These equations with the equations u1 = 0, ..., um = 0 are exactly enough to determine λ1, ..., λm, x1 x2, ..., xn, so that we find critical values of u, and examine the terms of the second order to decide whether we obtain a maximum or minimum.
To take a very simple illustration; consider the problem of determining the maximum and minimum radii vectors of the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, where a2 > b2 > c2. Here we require the maximum and minimum values of x2 + y2 + z2 where x2/a2 + y2/b2 + z2/c2 = 1.
We have
| δu = 2xδx ( 1 + | λ | ) + 2yδy ( | λ | ) + 2zδz ( | λ | ) |
| a2 | b2 | c2 |
| + δx2 ( 1 + | λ | ) + δy2 ( | λ | ) + δz2 ( | λ | ). |
| a2 | b2 | c2 |
To make the terms of the first order disappear, we have the three equations:—