Everything about Differential Infinitesimal totally explained
In
differential calculus, a
differential is traditionally an
infinitesimally small change in a
variable. For example, if
x is a variable, then a change in the value of
x is often denoted Δ
x (or
δx when this change is considered to be small). The differential d
x represents such a change, but is infinitely small. Although this isn't a rigorous mathematical concept, it's 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 d
y of
y is related to d
x by the formula
» . Furthermore, it has the decisive advantage over other definitions of the derivative that it's
invariant under changes of coordinates. This means that the same idea can be used to define the
differential of
smooth maps between
smooth manifolds.
Aside: Note that the existence of all the
partial derivatives of
at
x is a necessary condition for the existence of a differential at
x. However it isn't a sufficient condition. For counterexamples, see
Gateaux derivative.
The algebraic geometry approach
In
algebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the
coordinate ring or
sheaf of a space may contain
nilpotent elements. The simplest example is the ring of
dual numbers
R[
ε], where
ε2 = 0.
This can be motivated by the algebro-geometric point of view on the derivative of a function
f from
R to
R at a point
p. For this, note first that
f−
f(
p)
1 (where
1 is the identity function) belongs to the
ideal Ip of functions on
R which vanish at
p. If the derivative
f vanishes at
p, then
f−
f(
p)
1 belongs to the square
Ip2 of this ideal. Hence the derivative of
f at
p may be captured by the equivalence class [
f−
f(
p)
1] in the
quotient space Ip/
Ip2, and
the
1-jet of
f (which encodes its value and its first derivative) is the equivalence class of
f in the space of all functions modulo
Ip2. Algebraic geometers regard this equivalence class as the
restriction of
f to a
thickened version of the point
p whose coordinate ring isn't
R (which is the quotient space of functions on
R modulo
Ip) but
R[
ε] which is the quotient space of functions on
R modulo
Ip2. Such a thickened point is a simple example of a
scheme.
Synthetic differential geometry
A third approach to infinitesimals is the method of
synthetic differential geometry or
smooth infinitesimal analysis. This is closely related to the algebraic geometric approach, except that the infinitesimals are more implicit and intuitive. The main idea of this approach is to replace the
category of sets with another
category of
smoothly varying sets which is a
topos. In this category, one can define the real numbers, smooth functions, and so on, but the real numbers
automatically contain nilpotent infinitesimals, so these don't need to be introduced by hand as in the algebraic geometric approach. However the
logic in this new category isn't identical to the familiar logic of the category of sets: in particular, the
law of the excluded middle doesn't hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they're
constructive (for example, don't use
proof by contradiction). Some regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they're available.
Non-standard analysis
The final approach to infinitesimals again involves extending the real numbers, but in a much less drastic way. In this approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the
reciprocals of infinitely large numbers. Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of
rational numbers, so that, for example, the sequence (1,1/2,1/3,...1/n,...) represents an infinitesimal. The
first-order logic of this new set of
hyperreal numbers is the same as the logic for the usual real numbers, but the
completeness axiom (which involves
second-order logic) doesn't hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals.
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