There are several approaches for making the notion of differentials mathematically precise. • Differentials as
linear maps. This approach underlies the definition of the
derivative and the
exterior derivative in
differential geometry. • Differentials as
nilpotent elements of
commutative rings. This approach is popular in algebraic geometry. • 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. • Differentials as infinitesimals in
hyperreal number systems, which are extensions of the real numbers that 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 of being
quantitative, i.e., saying not just that a differential is infinitely small, but
how small it is.
Differentials as linear maps There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as
linear maps. It can be used on \mathbb{R}, \mathbb{R}^n, a
Hilbert space, a
Banach space, or more generally, a
topological vector space. The case of the Real line is the easiest to explain. This type of differential is also known as a
covariant vector or
cotangent vector, depending on context.
Differentials as linear maps on R Suppose f(x) is a real-valued function on \mathbb{R}. We can reinterpret the variable x in f(x) as being a function rather than a number, namely the
identity map on the real line, which takes a real number p to itself: x(p)=p. Then f(x) is the composite of f with x, whose value at p is f(x(p))=f(p). The differential \operatorname{d}f (which of course depends on f) is then a function whose value at p (usually denoted df_p) is not a number, but a linear map from \mathbb{R} to \mathbb{R}. Since a linear map from \mathbb{R} to \mathbb{R} is given by a 1\times 1
matrix, it is essentially the same thing as a number, but the change in the point of view allows us to think of df_p as an infinitesimal and
compare it with the
standard infinitesimal dx_p, which is again just the identity map from \mathbb{R} to \mathbb{R} (a 1\times 1
matrix with entry 1). The identity map has the property that if \varepsilon is very small, then dx_p(\varepsilon) is very small, which enables us to regard it as infinitesimal. The differential df_p has the same property, because it is just a multiple of dx_p, and this multiple is the derivative f'(p) by definition. We therefore obtain that df_p=f'(p)\,dx_p, and hence df=f'\,dx. Thus we recover the idea that f' is the ratio of the differentials df and dx. This would just be a trick were it not for the fact that: • it captures the idea of the derivative of f at p as the
best linear approximation to f at p; • it has many generalizations.
Differentials as linear maps on Rn If f is a function from \mathbb{R}^n to \mathbb{R}, then we say that f is
differentiable at p\in\mathbb{R}^n if there is a linear map df_p from \mathbb{R}^n to \mathbb{R} such that for any \varepsilon>0, there is a
neighbourhood N of p such that for x\in N, \left|f(x) - f(p) - df_p(x-p)\right| We can now use the same trick as in the one-dimensional case and think of the expression f(x_1, x_2, \ldots, x_n) as the composite of f with the standard coordinates x_1, x_2, \ldots, x_n on \mathbb{R}^n (so that x_j(p) is the j-th component of p\in\mathbb{R}^n). Then the differentials \left(dx_1\right)_p, \left(dx_2\right)_p, \ldots, \left(dx_n\right)_p at a point p form a
basis for the
vector space of linear maps from \mathbb{R}^n to \mathbb{R} and therefore, if f is differentiable at p, we can write as a
linear combination of these basis elements: df_p = \sum_{j=1}^n D_j f(p) \,(dx_j)_p. The coefficients D_j f(p) are (by definition) the
partial derivatives of f at p with respect to x_1, x_2, \ldots, x_n. Hence, if f is differentiable on all of \mathbb{R}^n, we can write, more concisely: df = \frac{\partial f}{\partial x_1} \,dx_1 + \frac{\partial f}{\partial x_2} \,dx_2 + \cdots +\frac{\partial f}{\partial x_n} \,dx_n. In the one-dimensional case this becomes df = \frac{df}{dx}dx as before. This idea generalizes straightforwardly to functions from \mathbb{R}^n to \mathbb{R}^m. Furthermore, it has the decisive advantage over other definitions of the derivative that it is
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 f(x) at x is a
necessary condition for the existence of a differential at x. However it is not a
sufficient condition. For counterexamples, see
Gateaux derivative.
Differentials as linear maps on a vector space The same procedure works on a vector space with enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as a
complete inner product space, where the inner product and its associated
norm define a suitable concept of distance. The same procedure works for a Banach space, also known as a complete
normed vector space. However, for a more general topological vector space, some of the details are more abstract because there is no concept of distance. For the important case of a finite dimension, any inner product space is a Hilbert space, any normed vector space is a Banach space and any topological vector space is complete. As a result, we can define a coordinate system from an arbitrary basis and use the same technique as for \mathbb{R}^n.
Differentials as germs of functions This approach works on any
differentiable manifold. If • and are open sets containing • f\colon U\to \mathbb{R} is continuous • g\colon V\to \mathbb{R} is continuous then is equivalent to at , denoted f \sim_p g, if and only if there is an open W \subseteq U \cap V containing such that f(x) = g(x) for every in . The germ of at , denoted [f]_p, is the set of all real continuous functions equivalent to at ; if is smooth at then [f]_p is a smooth germ. If • U_1, U_2 V_1 and V_2 are open sets containing • f_1\colon U_1\to \mathbb{R}, f_2\colon U_2\to \mathbb{R}, g_1\colon V_1\to \mathbb{R} and g_2\colon V_2\to \mathbb{R} are smooth functions • f_1 \sim_p g_1 • f_2 \sim_p g_2 • is a real number then • r*f_1 \sim_p r*g_1 • f_1+f_2\colon U_1 \cap U_2\to \mathbb{R} \sim_p g_1+g_2\colon V_1 \cap V_2\to \mathbb{R} • f_1*f_2\colon U_1 \cap U_2\to \mathbb{R} \sim_p g_1*g_2\colon V_1 \cap V_2\to \mathbb{R} This shows that the germs at p form an
algebra. Define \mathcal{I}_p to be the set of all smooth germs vanishing at and \mathcal{I}_p^2 to be the
product of
ideals \mathcal{I}_p \mathcal{I}_p. Then a differential at (cotangent vector at ) is an element of \mathcal{I}_p/\mathcal{I}_p^2. The differential of a smooth function at , denoted \mathrm d f_p, is [f-f(p)]_p/\mathcal{I}_p^2. A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch. Then the differential of at is the set of all functions differentially equivalent to f-f(p) at .
Algebraic geometry In
algebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the
coordinate ring or
structure 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) belongs to the
ideal Ip of functions on
R which vanish at
p. If the derivative
f vanishes at
p, then
f −
f(
p) 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)] 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 is not
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. 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 do not need to be introduced by hand as in the algebraic geometric approach. However the
logic in this new category is not identical to the familiar logic of the category of sets: in particular, the
law of the excluded middle does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are
constructive (e.g., do not use
proof by contradiction).
Constructivists regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.
Nonstandard analysis The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way. In the
nonstandard analysis 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
real 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) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see
transfer principle. ==Differential geometry==