Page 161 - Foundations Of Differential Calculus

144 8. On the Higher Differentiation of Differential Formulas
247. We let V be any finite function of x and y, and let dV = Pdx+Qdy.
In order to find the second differential of V we suppose that both of the
differentials dx and dy are variable. Since P and Q are functions of x and
y,welet

dP = pdx + rdy,
dQ = rdx + qdy,

since we have already noted that

∂P ∂Q
= = r.
∂y ∂x
Under these conditions we differentiate dV = Pdx + Qdy and obtain

2 2 2 2 2
d V = Pd x + pdx +2rdx dy + Qd y + qdy .
Hence, if we suppose that dx is constant, then

2 2 2 2
d V = pdx +2rdx dy + Qd y + qdy .
On the other hand, if we suppose that dy is constant, then
2 2 2 2
d V = Pd x + pdx +2rdx dy + qdy .

248. Therefore, if any function of x and y is differentiated twice, with
neither differential held constant, the second differential always has the
form
2 2 2 2 2
d V = Pd x + Qd y + Rdx + Sdy + Tdx dy;

where the quantities P, Q, R, S, and T are so interrelated that when we
use the notation used in the previous chapter,

∂P ∂Q ∂P ∂Q ∂Q ∂P
= , R = , S = , T =2 =2 .
∂y ∂x ∂x ∂y ∂x ∂y
If any one of these conditions fails, then we can affirm with certainty that
the proposed formula cannot be the second differential of a function. Here
we have an immediate test of whether or not an expression of this kind is
the second differential of some quantity.
249. In a similar way the third differential and higher differentials are
found. It would seem to be helpful to show a particular example rather
than give general formulas.