Page 142 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 125
213. Let us consider first a function of only two variables x and y, which
we will call V . Then its differential will have the form

dV = pdx + qdy.

If we let the quantity y remain constant, then dy = 0, so that the differential
of the function V will be pdx. If, on the other hand, we let x remain
constant, then dx = 0 and only y remains variable, so that the differential
of V is equal to qdy. The result is that the rule for differentiating a function
V of two variables x and y is as follows:
Let only the first quantity x remain variable, while the second quantity y
is treated as a constant. We take the differential of V , which will be equal
to pdx. Then we let only the quantity y remain variable, while the other,
x, is kept as constant and the differential is found, which will be equal to
qdy. From these results, when we let both x and y be variable we have
dV = pdx + qdy.
214. In a similar way, when the function V is of three variables x, y, and
z, the differential of this function has the form

dV = pdx + qdy + rdz.

It is clear that if only the quantity x is kept variable and the remaining y
and z are kept constant, since dy = 0 and dz = 0, the differential of V will
be equal to pdx. In a like manner we find the differential of V to be equal
to qdy when x and z are constant while only y is variable. If x and y are
treated as constants and only z is variable, we see that the differential of V
is equal to rdz. Hence, in order to find the differential of a function of three
or more variables, we consider individually each variable as if it alone were
variable and then take the differential, considering the other quantities as
constant. Then we take the sum of each of these differentials found with
each individual quantity taken as variable. This sum will be the required
differential of the given function.
215. In this rule, which we have found for the differentiation of a function
of however many variables, we have a demonstration of the general rule
given above (paragraph 170) by means of which any function of one variable
can be differentiated. If for each term of the function discussed in that
place the variable is considered to be a different letter and then each term
is successively differentiated in the way we have just prescribed, as if only
that were variable, we then collect into one sum each of the differentials
obtained in this way. This sum will be the desired differential after each
of the individual letters has its value restored. Hence this rule has wide
application to functions of several variables, no matter how they may be
composed. Thus its use in all of differential calculus is quite wide.