Differentials Of Various Orders For Functions Of Several Variables

Let $$u=f(x, y, z, \dots )$$ be a function of several independent variables x, y, z, $$ \dots . \ $$ If we differentiate this function several times in sequence, either with respect to all the variables or with respect to only one of them, we will obtain several new functions, each of which will be the total or partial derivative of the preceding one. We could also conceive that the successive differentiations are sometimes with respect to one variable, sometimes to another one. In all cases, the result of one, of two, of three, ... differentiations, successively performed, is what we call a total Total differential Total differential of various orders or partial differential Partial differential Partial differential of various orders of first, of second, of third, ... order. Thus, for example, by differentiating several times in sequence with respect to all the variables, we will generate the total differentials $$ du, ddu, dddu, \dots $$ that we denote, for brevity, by the notations $$ du, d^2u, d^3u, \dots . \ $$ On the other hand, by differentiating several times in sequence with respect to the variable x, we will generate the partial differentials $$ d_xu, d_x d_xu, d_x d_x d_xu, \dots $$ that we denote by the notations $$ d_xu, d_x^2u, d_x^3u, \dots . \ $$ In general, if n is any integer number, the total differential of order n will be represented by $$d^nu, $$ and the differential of the same order relative to only one of the variables $$ x, $$ y, z, $$ \dots $$ by $$ d_x^nu, d_y^nu, d_z^nu, \dots .$$ If we differentiate twice or several times in sequence with respect to two or to several variables, we would obtain the partial differentials of second order or of higher orders, designated by the notations $$ d_x d_yu, d_y d_xu, d_x d_zu, \dots , $$ $$ d_x d_y d_zu, $$ $$ \dots . \ $$ Now, it is easy to see that the differentials of this type retain the same values when we reverse the order in which the differentiations relative to the various variables must be performed.