Page 154 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 137
same expression, since there are six ways of ordering x, y, and z. No matter
what order is chosen, if the function V is differentiated with only the first
variable, and that is then differentiated with only the second variable, and
this then differentiated with only the third variable, the same expression is
obtained when the order is changed.

236. In order that the reason for this property may be seen more clearly,
we let

dV = Pdx + Qdy + Rdz.

Then we differentiate each of the quantities P, Q, and R, with their differ-
entials of the form we have already seen:

dP = pdx + sdy + tdz,
dQ = sdx + qdy + udz,
dR = tdx + udy + rdz.

Now if we differentiate V with only x variable, we have Pdx. This differ-
ential is now differentiated with only y variable to obtain sdxdy. If this
is differentiated with only z variable, and after this is divided by dx dy dz,
we have ∂s/∂z. Now we reorder the variables as y, x, z, and the first dif-
ferentiation gives Qdy, the second sdxdy and the third (when divided
by dx dy dz) gives ∂s/∂z as before. Now choose the order z, y, x and the
first differentiation gives Rdz, the second udy dz, and the third, after di-
vision by dx dy dz, gives ∂u/∂x. But when y is kept constant, we have
dQ = sdx + udz, so that

∂s ∂u
= ,
∂z ∂x

as we wished to show.

237. We let

2
x y
V = ,
2
a − z 2
and we take the three derivatives as many times as the order of the variables
x, y, and z can change: