Page 153 - Foundations Of Differential Calculus

136 7. On the Differentiation of Functions of Two or More Variables
so that

∂Q (n − 1) P px
= − ,
∂x y y
and we must have
(n − 1) P px
− = r,
y y
or

(n − 1) P = px + ry.

This equality becomes clear when we note that P is a homogeneous function
in x and y of dimension n − 1, so that its differential dP = pdx + rdy, due
to the property of homogeneous functions, should be such that (n − 1) P =
px + ry.

234. This property, that
∂P ∂Q
= ,
∂y ∂x
which we have shown to be common to all functions of two variables x and
y, can also reveal to us the nature of functions of three or more variables.
Let V be any function of three variables x, y, and z, and let
dV = Pdx + Qdy + Rdz.

If in this differentiation z is treated as a constant, then dV = Pdx+Qdy.In
this case by what has gone before it should be true that ∂P/∂y = ∂Q/∂x.
Then if y is supposed to be constant, we have dV = Pdx + Rdz, so that
∂P/∂z = ∂R/∂x. Finally with x constant we see that ∂Q/∂z = ∂R/∂y.
Therefore, in the differential Pdx + Qdy + Rdz of the function V , the
quantities P, Q, and R are related to each other in such a way that
∂P ∂Q ∂P ∂R ∂Q ∂R
= , = , = .
∂y ∂x ∂z ∂x ∂z ∂y

235. It follows that this property of functions that involve three or more
variables is analogous to that which we have shown above (paragraph 230)
for functions of two variables. If V is any function of three variables x,
y, and z, and this is differentiated three times in such a way that in the
first differentiation the first variable, that is, x, is the only variable, in the
second differentiation only y is variable, and in the third only z is variable,
then we obtain an expression of the form Zdx dy dz. This same expression
is obtained no matter in which order the quantities x, y, and z are taken.
There are six different ways of taking the threefold derivative to obtain the