Differential (infinitesimal)

Differential (infinitesimal)

In calculus, a differential is traditionally an infinitesimal (infinitely small) change in a variable. For example, if x is a variable, then a change in the value of x is often denoted Δx (or δx when this change is considered to be small). The differential dx represents such a change, but is infinitely small. Although, as stated, it is not a precise mathematical concept, it is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.

The key property of the differential is that if y is a function of x, then the differential dy of y is related to dx by the formula

where dy/dx denotes the derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.

There are several approaches for making the notion of differentials mathematically precise.

1. Differentials as linear maps. This approach underlies the definition of the derivative and the exterior derivative in differential geometry.
2. Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry.
3. Differentials in smooth models of set theory. This approach is known as synthetic differential geometry or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced.
4. Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers which contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson.

These approaches are very different from each other, but they have in common the idea to be quantitative, i.e., to say not just that a differential is infinitely small, but how small it is.