Page 156 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 139
which the triple differentiation arrives at the same expression Zdx dy dz.
There are twenty-four ways of taking four derivatives to arrive at the same
expression of the form Zdx dy dz dv, and so forth.
239. One can easily agree to the truths of these theorems with little at-
tention paid to the principles explained earlier, and one can more easily see
this truth by one’s own meditation than by such complications of words,
without which it is not possible to give a demonstration. But since a knowl-
edge of these properties is of the greatest importance in integral calculus,
beginners should be warned that they must not only meditate on these
properties with great care and examine their truth, but also work through
many examples. In this way they will become very familiar with this ma-
terial, and then they will be able more easily to gather the fruit which
will come later. Indeed, not only beginners, but also those who are already
acquainted with the principles of differential calculus are exhorted to the
same, since in almost all introductions to this part of analysis this argument
is wont to be omitted. Frequently, authors have been content to give only
the rules for differentiation and the applications to higher geometry. They
do not inquire into the nature or the properties of differentials, from which
the greatest aid to integral calculus comes. For this reason the argument,
which is practically new in this chapter, has been discussed at length in
order that the way to other more difficult integrations may be prepared,
and the work to be undertaken later might be lightened.
240. Once we know these properties that functions of two or more vari-
ables enjoy, we can easily decide whether or not a given formula for a
differential in which there occur two or more variables has really arisen
from differentiation of some finite function. If in the formula Pdx + Qdy
it is not true that ∂P/∂y = ∂Q/∂x, then we can with certainty state that
there is no function of x and y whose differential is equal to Pdx + Qdy.
When we come to integral calculus we will deny that there is any integral
2
for such a formula. Hence since yx dx + x dy does not have the required
2
condition, there is no function whose differential is equal to yx dx + x dy.
The question is whether as long as ∂P/∂y = ∂Q/∂x, the formula has al-
ways arisen from the differentiation of some function. From the principles
of integration we can surely answer in the affirmative.
241. If in a given formula of a differential there are three or more variables,
such as Pdx+Qdy +Rdz, then there is no way that this shall have arisen
from differentiation unless these three conditions are met:

∂P ∂Q ∂P ∂R ∂Q ∂R
= , = , = .
∂y ∂x ∂z ∂x ∂z ∂y

Of these conditions, if even only one is missing, then we can state with