The sign of this expression in general is that of [Sigma]([delta]u/[delta]x1)[delta]x1, which cannot be one-signed when x1, x2, ... x_n can take all possible values, for a set of increments [delta]x1, [delta]x2 ... [delta]x_n, will give an opposite sign to the set -[delta]x1, -[delta]x2, ... -[delta]x_n. Hence [Sigma]([delta]u/[delta]x1)[delta]x1 must vanish for all sets of increments [delta]x1, ... [delta]x_n, and since these are independent, we must have [delta]u/[delta]x1 = 0, [delta]u/[delta]x2 = 0, ... [delta]u/[delta]x_n = 0. A value of u given by a set of solutions of these equations is called a "critical value" of u. The value of [delta]u now becomes _ _ | __ [delta]²u __ [delta]²u | ½ | \ --------- [delta]x1² + 2 \ ------------------- [delta]x1 [delta]x2 + ... |; |_ /__ [delta]x1² /__ [delta]x1 [delta]x2 _| for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d²u/dx² is negative or positive. If d²u/dx² vanishes, then there is no maximum or minimum unless d²u/dx² vanishes, and there is a maximum or minimum according as d^4u/dx^4 is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum. In the case of several variables, the quadratic __ [delta]²u __ [delta]²u \ ---------- [delta]x1² + 2 \ ------------------- + ... /__ [delta]x1² /__ [delta]x1 [delta]x2 must be one-signed. The condition for this is that the series of discriminants a11 , | a11 a12 | , | a11 a12 a13 | , ... | a21 a22 | | a21 a22 a23 | | a31 a32 a33 | where a_pq denotes [delta]²u/[delta]a_p[delta]a_q should be all positive, if the quadratic is always positive, and alternately negative and positive, if the quadratic is always negative. If the first condition is satisfied the critical value is a minimum, if the second it is a maximum. For the case of two variables the conditions are [delta]²u [delta]²u / [delta]² \² ---------- · ---------- > ( ------------------- ) [delta]x1² [delta]x2² \ [delta]x1 [delta]x2 / for a maximum or minimum at all and [delta]²u/[delta]x1² and [delta]²u/[delta]x2² both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when [delta]x1, [delta]x2, ... [delta]x_n all vanish. If, in the case of two variables,