Editing Differential (mathematics) (section)

== Approaches ==
{{Calculus |Differential}}

There are several approaches for making the notion of differentials mathematically precise.
# Differentials as [[linear map]]s. This approach underlies the definition of the [[total derivative|derivative]] and the [[exterior derivative]] in [[differential geometry]].<ref>{{Harvnb|Darling|1994}}.</ref>
# Differentials as [[nilpotent]] elements of [[commutative ring]]s. This approach is popular in algebraic geometry.<ref name="Harris1998">{{Harvnb|Eisenbud|Harris|1998}}.</ref>
# 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.<ref>See {{Harvnb|Kock|2006}} and {{Harvnb|Moerdijk|Reyes|1991}}.</ref>
# 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]].<ref name="nonstd">See {{Harvnb|Robinson|1996}} and {{Harvnb|Keisler|1986}}.</ref>

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 map]]s. It can be used on <math>\mathbb{R}</math>, <math>\mathbb{R}^n</math>, 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 <math>f(x)</math> is a real-valued function on <math>\mathbb{R}</math>.  We can reinterpret the variable <math>x</math> in <math>f(x)</math> as being a function rather than a number, namely the [[identity map]] on the real line, which takes a real number <math>p</math> to itself: <math>x(p)=p</math>.  Then <math>f(x)</math> is the composite of <math>f</math> with <math>x</math>, whose value at <math>p</math> is <math>f(x(p))=f(p)</math>. The differential <math>\operatorname{d}f</math> (which of course depends on <math>f</math>) is then a function whose value at <math>p</math> (usually denoted <math>df_p</math>) is not a number, but a linear map from <math>\mathbb{R}</math> to <math>\mathbb{R}</math>. Since a linear map from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> is given by a <math>1\times 1</math> [[Matrix (mathematics)|matrix]], it is essentially the same thing as a number, but the change in the point of view allows us to think of <math>df_p</math> as an infinitesimal and ''compare'' it with the ''standard infinitesimal'' <math>dx_p</math>, which is again just the identity map from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> (a <math>1\times 1</math> [[Matrix (mathematics)|matrix]] with entry <math>1</math>). The identity map has the property that if <math>\varepsilon</math> is very small, then <math>dx_p(\varepsilon)</math> is very small, which enables us to regard it as infinitesimal. The differential <math>df_p</math> has the same property, because it is just a multiple of <math>dx_p</math>, and this multiple is the derivative <math>f'(p)</math> by definition. We therefore obtain that <math>df_p=f'(p)\,dx_p</math>, and hence <math>df=f'\,dx</math>. Thus we recover the idea that <math>f'</math> is the ratio of the differentials <math>df</math> and <math>dx</math>.

This would just be a trick were it not for the fact that:
# it captures the idea of the derivative of <math>f</math> at <math>p</math> as the ''best linear approximation'' to <math>f</math> at <math>p</math>;
# it has many generalizations.

==== Differentials as linear maps on R<sup>n</sup> ====

If <math>f</math> is a function from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math>, then we say that <math>f</math> is ''differentiable''<ref>See, for instance, {{Harvnb|Apostol|1967}}.</ref> at <math>p\in\mathbb{R}^n</math> if there is a linear map <math>df_p</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> such that for any <math>\varepsilon>0</math>, there is a [[neighbourhood (mathematics)|neighbourhood]] <math>N</math> of <math>p</math> such that for <math>x\in N</math>,
<math display=block>\left|f(x) - f(p) - df_p(x-p)\right| < \varepsilon \left|x-p\right| .</math>

We can now use the same trick as in the one-dimensional case and think of the expression <math>f(x_1, x_2, \ldots, x_n)</math> as the composite of <math>f</math> with the standard coordinates <math>x_1, x_2, \ldots, x_n</math> on <math>\mathbb{R}^n</math> (so that <math>x_j(p)</math> is the <math>j</math>-th component of <math>p\in\mathbb{R}^n</math>). Then the differentials <math>\left(dx_1\right)_p, \left(dx_2\right)_p, \ldots, \left(dx_n\right)_p</math> at a point <math>p</math> form a [[basis (linear algebra)|basis]] for the [[vector space]] of linear maps from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> and therefore, if <math>f</math> is differentiable at <math>p</math>, we can write ''<math>\operatorname{d}f_p</math>'' as a [[linear combination]] of these basis elements:
<math display=block>df_p = \sum_{j=1}^n D_j f(p) \,(dx_j)_p.</math>

The coefficients <math>D_j f(p)</math> are (by definition) the [[partial derivative]]s of <math>f</math> at <math>p</math> with respect to <math>x_1, x_2, \ldots, x_n</math>. Hence, if <math>f</math> is differentiable on all of <math>\mathbb{R}^n</math>, we can write, more concisely:
<math display=block>\operatorname{d}f = \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.</math>

In the one-dimensional case this becomes
<math display=block>df = \frac{df}{dx}dx</math>
as before.

This idea generalizes straightforwardly to functions from <math>\mathbb{R}^n</math> to <math>\mathbb{R}^m</math>. Furthermore, it has the decisive advantage over other definitions of the derivative that it is [[invariant (mathematics)|invariant]] under changes of coordinates. This means that the same idea can be used to define the [[pushforward (differential)|differential]] of [[smooth map]]s between [[smooth manifold]]s.

Aside: Note that the existence of all the [[partial derivative]]s of <math>f(x)</math> at <math>x</math> is a [[necessary condition]] for the existence of a differential at <math>x</math>. 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 a enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as a [[Complete metric space|complete]] [[inner product space]], where the inner product and its associated [[Norm (mathematics)|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, you can define a coordinate system from an arbitrary basis and use the same technique as for <math>\mathbb{R}^n</math>.

=== Differentials as germs of functions ===

This approach works on any [[differentiable manifold]]. If
# {{var|U}} and {{var|V}} are open sets containing {{var|p}}
# <math>f\colon U\to \mathbb{R}</math> is continuous
# <math>g\colon V\to \mathbb{R}</math> is continuous
then {{var|f}} is equivalent to {{var|g}} at {{var|p}}, denoted <math>f \sim_p g</math>, if and only if
there is an open <math>W \subseteq U \cap V</math> containing {{var|p}} such that <math>f(x) = g(x)</math> for every {{var|x}} in {{var|W}}.
The germ of {{var|f}} at {{var|p}}, denoted <math>[f]_p</math>, is the set of all real continuous functions equivalent to {{var|f}} at {{var|p}}; if {{var|f}} is smooth at {{var|p}} then <math>[f]_p</math> is a smooth germ.
If
#<math>U_1</math>, <math>U_2</math> <math>V_1</math> and <math>V_2</math> are open sets containing {{var|p}}
#<math>f_1\colon U_1\to \mathbb{R}</math>, <math>f_2\colon U_2\to \mathbb{R}</math>, <math>g_1\colon V_1\to \mathbb{R}</math> and <math>g_2\colon V_2\to \mathbb{R}</math> are smooth functions
#<math>f_1 \sim_p g_1</math>
#<math>f_2 \sim_p g_2</math>
#{{var|r}} is a real number
then
#<math>r*f_1 \sim_p r*g_1</math>
#<math>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}</math>
#<math>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}</math>
This shows that the germs at p form an [[Algebra over a field|algebra]].

Define <math>\mathcal{I}_p</math> to be the set of all smooth germs vanishing at {{var|p}} and
<math>\mathcal{I}_p^2</math> to be the [[Ideal (ring theory)#Ideal operations|product]] of [[Ideal (ring theory)|ideals]] <math>\mathcal{I}_p \mathcal{I}_p</math>. Then a differential at {{var|p}} (cotangent vector at {{var|p}}) is an element of <math>\mathcal{I}_p/\mathcal{I}_p^2</math>. The differential of a smooth function {{var|f}} at {{var|p}}, denoted <math>\mathrm d f_p</math>, is <math>[f-f(p)]_p/\mathcal{I}_p^2</math>.

A similar approach is to define differential equivalence of first order in terms of derivatives in an arbitrary coordinate patch.
Then the differential of {{var|f}} at {{var|p}} is the set of all functions differentially equivalent to <math>f-f(p)</math> at {{var|p}}.

=== 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 element]]s. The simplest example is the ring of [[dual number]]s '''R'''[''ε''], where ''ε''<sup>2</sup> = 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''&nbsp;&minus; ''f''(''p'') belongs to the [[ideal (ring theory)|ideal]] ''I''<sub>''p''</sub> of functions on '''R''' which vanish at ''p''. If the derivative ''f'' vanishes at ''p'', then ''f''&nbsp;&minus; ''f''(''p'') belongs to the square ''I''<sub>''p''</sub><sup>2</sup> of this ideal. Hence the derivative of ''f'' at ''p'' may be captured by the equivalence class [''f''&nbsp;&minus; ''f''(''p'')] in the [[quotient space (linear algebra)|quotient space]]  ''I''<sub>''p''</sub>/''I''<sub>''p''</sub><sup>2</sup>, and the [[jet (mathematics)|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 ''I''<sub>''p''</sub><sup>2</sup>. 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 ''I''<sub>''p''</sub>) but '''R'''[''ε''] which is the quotient space of functions on '''R''' modulo ''I''<sub>''p''</sub><sup>2</sup>. Such a thickened point is a simple example of a [[Scheme (mathematics)|scheme]].<ref name="Harris1998" />

==== Algebraic geometry notions ====
<!-- Integrate text. -->
Differentials are also important in [[algebraic geometry]], and there are several important notions.
* [[Abelian differential]]s usually mean differential one-forms on an [[algebraic curve]] or [[Riemann surface]].
* [[Quadratic differential]]s (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
* [[Kähler differential]]s provide a general notion of differential in algebraic geometry.

=== Synthetic differential geometry ===

A fifth approach to infinitesimals is the method of [[synthetic differential geometry]]<ref>See {{Harvnb|Kock|2006}} and {{Harvnb|Lawvere|1968}}.</ref> or [[smooth infinitesimal analysis]].<ref>See {{Harvnb|Moerdijk|Reyes|1991}} and {{Harvnb|Bell|1998}}.</ref> 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 (mathematics)|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 mathematics|constructive]]'' (e.g., do not use [[proof by contradiction]]). [[Constructivism_(philosophy_of_mathematics)|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 [[multiplicative inverse|reciprocal]]s of infinitely large numbers.<ref name="nonstd"/> Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of [[real number]]s, so that, for example, the sequence (1,&nbsp;1/2,&nbsp;1/3,&nbsp;...,&nbsp;1/''n'',&nbsp;...) represents an infinitesimal. The [[first-order logic]] of this new set of [[hyperreal number]]s 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]].