CONDITIONS WHICH MUST BE FULFILLED FOR A TOTAL DIFFERENTIAL TO NOT CHANGE SIGN WHILE WE CHANGE THE VALUES ATTRIBUTED TO THE DIFFERENTIALS OF THE INDEPENDENT VARIABLES.

• Dennis M. Cates
Chapter

Abstract

After what we have seen in the preceding lectures, if we denote by u a function of the independent variables $$x, y, z, \dots ,$$ and if we disregard the values of these variables which render one of the functions $$u, du, d^2u, \dots$$ discontinuous, the function u can only become a maximum or a minimum in the case where one of the total differentials $$d^2u,$$ $$d^4u,$$ $$d^6u,$$ $$\dots ,$$ namely, the first of these that will not be constantly null, will maintain the same sign for all possible values of the arbitrary quantities $$dx=h, dy=k, dz=l, \dots ,$$ or at least for the values of these quantities which will not reduce it to zero. Add that, in the latter assumption, each of the systems of values of $$h, k, l, \dots$$ that work to make the total differential in question vanish, must change another total differential of even order into a quantity affected by the sign that maintains the first differential, as long as it does not vanish.