Page 148 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 131

2 x + y
IV. If V = x ln , then n = 2 and
y − x
y + x 2x (ydx − xdy)
2
dV =2xdx ln + .
2
y − x y − x 2
When we make the required substitutions, there arises

2 y + x
2x ln =2V.
y − x

224. A similar property can be observed if V is a homogeneous function
in more than two variables. If V is a function of the quantities x, y, and
z, which together have a dimension of n, then its differential has the form
Pdx+Qdy+Rdz. When we let y = tx and z = sx, so that dy = tdx+xdt
and dz = sdx + xds, then furthermore, the function V takes the form
n
Ux , where U is a function of the two variables t and s. Hence, if we let
dU = pdt + qds, then
n
n
dV = x pdt + x qds + nUx n−1 dx.
The previous form gives
dV = Pdx + Qt dx + Qx dt + Rs dx + Rx ds.

When these two forms are compared, we have

n−1 nV
P + Qt + Rs = nUx = ,
x
from which we obtain

Px + Qy + Rz = nV.

This same property extends to functions of howsoever many variables.
225. If the given function is homogeneous in howsoever many variables
x,y,z,v,... , its differential will always have the property that, if for the dif-
ferentials dx,dy,dz,dv,... we substitute the finite quantities x,y,z,v,... ,
then the result is the given function multiplied by the dimension. This rule
applies likewise if V is a homogeneous function of the single variable x.In
n
this case V isapowerof x, for example V = ax , which is a homogeneous
function of dimension n. Indeed, there is no other function of x in which
n
x has n dimensions besides the power x . Since dV = nax n−1 dx, when we
n
substitute x for dx, we obtain nax , which is nV . This remarkable property
of homogeneous functions deserves to be very carefully noted, since it will
have extremely useful consequences in integral calculus.