Page 151 - Foundations Of Differential Calculus

134 7. On the Differentiation of Functions of Two or More Variables
III. Let V = x sin y + y sin x, so that

dV = dx sin y + xdy cos y + dy sin x + ydx cos x.
Hence Pdx = dx sin y + ydx cos x and Qdy = dy sin x + xdy cos y.
When we keep x constant, we have
d.P dx = dx dy cos y + dx dy cos x,

and when y is kept constant, we have
d.Q dy = dx dy cos y + dx dy cos x.

y
IV. Let V = x . Then
y
dV = x dy ln x + yx y−1 dx,
y
so that Pdx = yx y−1 dx and Qdy = x dy ln x. Hence when we keep
x constant we have
y−1 y−1
d.P dx = x dx dy + yx dx dy ln x,
and when y is kept constant, we have
y−1 y−1
d.Q dy = yx dx dy ln x + x dx dy.

230. This property can also be stated in another way, so that this remark-
able characteristic of all functions of two variables can be understood. If
any function V of two variables x and y is differentiated with only x vari-
able, and this differential is again differentiated with only y variable, then
after this double differentiation, the same result is obtained when the order
of differentiation is reversed by first differentiating V with only y variable
and then differentiating this differential with only x variable. Both of these
cases give the same expression of the form zdx dy. The reason for this
identity clearly follows from the previous property; for if V is differentiated
with only x variable, we have Pdx, and if V is differentiated with only y
variable, we have Qdy. The differentials of these, in the way already indi-
cated, are equal, as we have demonstrated. For the rest, this characteristic
follows immediately from the argument given in paragraph 227.

231. The relationship between P and Q,if Pdx + Qdy is the differential
of the function V , can also be indicated in the following way. Since P and Q
are functions of x and y, they can both be differentiated with both x and y
variable. If dV = Pdx + Qdy, then dP = pdx + rdy and dQ = qdx + sdy.
Therefore, when x is constant, dP = rdy and d.P dx = rdx dy. Then when
y is constant dQ = qdx and d.Q dy = qdx dy. Since these two differentials
rdx dy and qdx dy are equal to each other, it follows that
q = r.