Thus the differential of an independent variable such as x is a finite difference; in other words it is any number we please. The differential of a dependent variable such as y, or of a function of the independent variable x, is the product of the differential of x and the differential coefficient or derived function. It is important to observe that the differential coefficient is not to be defined as the ratio of differentials, but the ratio of differentials is to be defined as the previously introduced differential coefficient. The differentials are either finite differences, or are so much of certain finite differences as are useful for forming differential coefficients.
Again let F(x) be the indefinite integral of a continuous function ƒ(x), so that we have
| dF(x) | = ƒ(x), ∫ba ƒ(x) dx = F(b) − F(a). |
| dx |
When the points M of the process explained in § 5 are inserted between the points whose abscissae are a and b, we may take them to be n − 1 in number, so that the segment AB is divided into n segments. Let x1, x2, ... xn−1 be the abscissae of the points in order. The integral is the limit of the sum
ƒ(a) (x1 − a) + ƒ(x1) (x2 − x1) + ... + ƒ(xr) (xr+1 − xr) + ... + ƒ(xn−1) (b − xn−1),
every term of which is a differential of the form ƒ(x)dx. Further the integral is equal to the sum of differences
{F(x1) − F(a)} + {F(x2) − F(x1)} + ... + {F(xr+1) − F(xr)} + ... + {F(b) − F(xn−1)},
for this sum is F(b) − F(a). Now the difference F(xr+1) − F(xr) is not equal to the differential ƒ(xr) (xr+1 − xr), but the sum of the differences is equal to the limit of the sum of these differentials. The differential may be regarded as so much of the difference as is required to form the integral. From this point of view a differential is called a differential element of an integral, and the integral is the limit of the sum of differential elements. In like manner the differential element ydx of the area of a curve (§ 5) is not the area of the portion contained between two ordinates, however near together, but is so much of this area as need be retained for the purpose of finding the area of the curve by the limiting process described.
8. The notation of the infinitesimal calculus is intimately bound up with the notions of differentials and sums of elements. The letter Notation.
Fundamental Artifice. “d” is the initial letter of the word differentia (difference) and the symbol ∫ is a conventionally written “S,” the initial letter of the word summa (sum or whole). The notation was introduced by Leibnitz (see §§ 25-27, below).
9. The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x + Δx for x and y + Δy for y. If there is a differential coefficient of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx=0 (R/Δx) = 0, as in § 7 above. The artifice consists in rejecting ab initio all terms of the equation which belong to R. We do not form R at all, but only φ·Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form the element of an integral in the same way as the element of area y·dx is formed. In fig. 3 of § 5 the element of area y·dx is the area of the rectangle RM. The actual area of the curvilinear figure PQNM is greater than the area of this rectangle by the area of the curvilinear figure PQR; but the excess is less than the area of the rectangle PRQS, which is measured by the product of the numerical measures of MN and QR, and we have