微分小のソースを表示

{{otheruses||その他の用法|微分 (曖昧さ回避)}}
{{Calculus |Differential}}
[[初等解析学]]（[[微分積分学]]）において{{要出典範囲|'''微分小'''（びぶんしょう|date=2016年11月}}、{{lang-en-short|''differential''}}）の語は、適当な[[変数 (数学)|変量]]に関する[[無限小]]変分を指すために用いられる。例えば、変数 {{mvar|x}} に対してその増分（変分）はしばしば {{mvar|Δx}} と書かれるが、変数 {{mvar|x}} に関する無限に小さな増分を表すのに {{mvar|dx}} が用いられる。無限小変分（'''微分小'''）の概念は直観的な議論においてきわめて有効であり、またその数学的に意味のある定式化にはいくつもの方法が存在する。

初等解析学において、さまざまな変数に関する無限小変分の間の関係性を[[微分商]]を用いて述べることができる。{{mvar|y}} が {{mvar|x}} の函数であるとき、{{mvar|y}} の微分 {{mvar|dy}} は {{mvar|dx}} との間に等式
: <math> \mathit{dy} = \frac{\mathit{dy}}{\mathit{dx}} \,\mathit{dx}</math>
を通じて関係を持つ。ここに {{mvar|{{fraction|dy|dx}}}} は {{mvar|y}} の {{mvar|x}} に関する微分商である。 この式は「{{mvar|x}} に関する {{mvar|y}} の微分商とは差分商 {{mvar|{{fraction|Δy|Δx}}}} の {{mvar|Δx}} を無限小に近づけた極限である」という直観的な考えをまとめたものである。

微分小量の概念を数学的に明確にする方法には、例えば以下のようなものが考えられる：
# [[線型写像]]として: これは[[全微分]]および[[微分幾何学]]における[[外微分]]の定義を下敷きにしたものである<ref>{{Harvnb|Darling|1994}}.</ref>。
# [[可換環]]の[[冪零元]]として: この方法は代数幾何学ではよく用いられる<ref name="Harris1998">{{Harvnb|Eisenbud|Harris|1998}}.</ref>。
# 直観主義論理の枠組みで: この方法は{{仮リンク|綜合微分幾何学|en|synthetic differential geometry}}や[[滑らかな無限小解析]]といわれるもので、冪零無限小が導入されるという点では代数幾何学的な方法と近いが、そうなるメカニズムは全く異なり[[トポス理論]]からくる<ref>See {{Harvnb|Kock|2006}} and {{Harvnb|Moerdijk|Reyes|1991}}.</ref>。
# [[超実数]]の無限小元として: 超実数は可逆な無限小や無限大を含むような実数概念の拡張である。このような方法は[[アブラハム・ロビンソン]]の開拓した[[超準解析]]による<ref name="nonstd">See {{Harvnb|Robinson|1996}} and {{Harvnb|Keisler|1986}}.</ref>。

これらのアプローチの各々は互いに非常に異なっているけれども、いずれも「定量的」な概念であることは共通している。つまりこれらの方法で定式化された微分は「無限に小さい」のではなく「どれほどでも（必要なだけ十分に）小さい」のである。

== 歴史と用例 ==
{{See also|{{仮リンク|微分積分学の歴史|en|History of calculus}}}}
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[[Infinitesimal]] quantities played a significant role in the development of calculus. [[Archimedes]] used them, even though he didn't believe that arguments involving infinitesimals were rigorous.<ref>{{Harvnb|Boyer|1991}}.</ref> [[Isaac Newton]] referred to them as [[Method of Fluxions|fluxions]]. However, it was [[Gottfried Leibniz]] who coined the term ''differentials'' for infinitesimal quantities, and introduced the notation for them which is still used today.

In [[Leibniz's notation]], if ''x'' is a variable quantity, then d''x'' denotes an infinitesimal change in the variable ''x''. Thus, if ''y'' is a function of ''x'', then the [[derivative]] of ''y'' with respect to ''x'' is often denoted d''y''/d''x'', which would otherwise be denoted (in the notation of Newton or Lagrange) ''ẏ'' or ''y''&thinsp;&prime;. The use of differentials in this form attracted much criticism, for instance in the famous pamphlet [[The Analyst]] by Bishop Berkeley. Nevertheless the notation has remained popular because it suggests strongly the idea that the derivative of ''y'' at ''x'' is its [[instantaneous rate of change]] (the [[slope (mathematics)|slope]] of the graph's [[tangent line]]), which may be obtained by taking the [[limit (mathematics)|limit]] of the ratio Δ''y''/Δ''x'' of the change in ''y'' over the change in ''x'', as the change in ''x'' becomes arbitrarily small. Differentials are also compatible with [[dimensional analysis]], where a differential such as d''x'' has the same dimensions as the variable ''x''.

Differentials are also used in the notation for [[integral]]s because an integral can be regarded as an infinite sum of infinitesimal quantities: the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as
: <math> \int f(x) \,\mathrm{d}x ,\,</math>
the integral sign (which is a modified [[long s]]) denotes the infinite sum, ''f''(''x'') denotes the "height" of a thin strip, and the differential d''x'' denotes its infinitely thin width.
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== 線型主要部 ==
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There is a simple way to make precise sense of differentials by regarding them as [[linear map]]s. One way to explain this point of view is to regard the variable ''x'' in an expression such as ''f''(''x'') as a function on the real line, the ''standard [[coordinate]]'' or [[identity map]], which takes a real number ''p'' to itself (''x''(''p'')&nbsp;= ''p''): then ''f''(''x'') denotes the composite ''f''&nbsp;∘ ''x'' of ''f'' with ''x'', whose value at ''p'' is ''f''(''x''(''p''))&nbsp;= ''f''(''p''). The differential d''f'' is then a function of ''f'' whose value at ''p'' (usually denoted d''f''<sub>''p''</sub>) is not a number, but a linear map from '''R''' to '''R'''. Since a linear map from '''R''' to '''R''' is given by a 1&times;1 [[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 d''f''<sub>''p''</sub> as an infinitesimal and ''compare'' it with the ''standard infinitesimal'' d''x''<sub>''p''</sub>, which is again just the identity map from '''R''' to '''R''' (a 1&times;1 [[Matrix (mathematics)|matrix]] with entry 1). The identity map has the property that if ε is very small, then d''x''<sub>''p''</sub>(ε) is very small which enables us to regard it as infinitesimal. The differential d''f''<sub>''p''</sub> has the same property, because it is just a multiple of d''x''<sub>''p''</sub>, and this multiple is the derivative ''f''&thinsp;&prime;(''p'') by definition. We therefore obtain that d''f''<sub>''p''</sub>&nbsp;= ''f''&thinsp;&prime;(''p'')&thinsp;d''x''<sub>''p''</sub>, and hence d''f''&nbsp;= ''f''&thinsp;&prime;&thinsp;d''x''. Thus we recover the idea that ''f''&thinsp;&prime; is the ratio of the differentials d''f'' and d''x''.

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.

For instance if ''f'' is a function from '''R'''<sup>''n''</sup> to '''R''' then we say ''f'' is ''differentiable''<ref>See, for instance, {{Harvnb|Apostol|1967}}.</ref> at ''p''&nbsp;∈ '''R'''<sup>''n''</sup> if there is a linear map d''f''<sub>''p''</sub> from '''R'''<sup>''n''</sup> to '''R''' such that for any ε&nbsp;> 0, there is a [[neighbourhood (mathematics)|neighbourhood]] ''N'' of ''p'' such that for ''x&nbsp;∈ ''N'':
: <math> \left|f(x) - f(p) - \mathrm{d}f_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 ''f''(''x''<sup>1</sup>, ''x''<sup>2</sup>, …, ''x''<sup>''n''</sup>) as the composite of ''f'' with the standard coordinates ''x''<sup>1</sup>, ''x''<sup>2</sup>, …, and ''x''<sup>''n''</sup> on '''R'''<sup>''n''</sup> (so that ''x''<sup>''j''</sup>(''p'') is the ''j''-th component of ''p''&nbsp;∈ '''R'''<sup>''n''</sup>). Then the differentials (d''x''<sup>1</sup>)<sub>''p''</sub>, (d''x''<sup>2</sup>)<sub>''p''</sub>, (d''x''<sup>''n''</sup>)<sub>''p''</sub> (at a point ''p'') form a [[basis (linear algebra)|basis]] for the [[vector space]] of linear maps from '''R'''<sup>''n''</sup> to '''R''' and therefore, if ''f'' is differentiable at ''p'', we can write d''f''<sub>''p''</sub> as a [[linear combination]] of these basis elements:
: <math> \mathrm{d}f_p = \sum_{j=1}^n D_j f(p) \,(\mathrm{d}x^{j})_p .\,</math>

The coefficients ''D''<sub>''j''</sub>''f''(''p'') are (by definition) the [[partial derivative]]s of ''f'' at ''p'' with respect to ''x''<sup>1</sup>, ''x''<sup>2</sup>, …, and ''x''<sup>''n''</sup>. Hence, if ''f'' is differentiable on all of '''R'''<sup>''n''</sup>, we can write, more concisely:
: <math> \mathrm{d}f = \frac{\partial f}{\partial x^1} \,\mathrm{d}x^1 + \frac{\partial f}{\partial x^2} \,\mathrm{d}x^2 + \cdots +\frac{\partial f}{\partial x^n} \,\mathrm{d}x^n. </math>

In the one-dimensional case this becomes
: <math> \mathrm{d}f = \frac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x </math>
as before.

This idea generalizes straightforwardly to functions from '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup>. 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 ''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 [[Gâteaux derivative]].
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== 代数幾何学 ==
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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'')'''1''' (where '''1''' is the identity function) 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'')'''1''' 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'')'''1'''] 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" />
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== 綜合微分幾何学 ==
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A third 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]]). Some{{who|date=November 2012}} regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.
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== 超準解析 ==
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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]].
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== 注 ==
{{脚注ヘルプ}}
=== 注釈 ===
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=== 出典 ===
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== 参考文献 ==
* {{Citation| first=Tom M.|last= Apostol|authorlink=Tom M. Apostol|year=1967|title=Calculus|edition= 2nd|publisher= Wiley|isbn=978-0-471-00005-1}}.
* {{Citation | last=[[John Lane Bell|Bell]]|first=John L.|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title= Invitation to Smooth Infinitesimal Analysis|year=1998}}.
* {{Citation |first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=978-0-471-54397-8|chapter=Archimedes of Syracuse}}.
* {{Citation |last=Darling|first= R. W. R. |title=Differential forms and connections |publisher=Cambridge University Press |location=Cambridge, UK |year=1994 |isbn=978-0-521-46800-8}}.
* {{Citation| author1-link=David Eisenbud|first1=David|last1=Eisenbud|author2-link=Joe Harris (mathematician)|first2=Joe|last2=Harris| year = 1998 |title = The Geometry of Schemes| publisher = Springer-Verlag| isbn =978-0-387-98637-1}}
* {{Citation |author-link=Howard Jerome Keisler|first=H. Jerome|last=Keisler|title=Elementary Calculus: An Infinitesimal Approach|edition=2nd|year=1986|url=http://www.math.wisc.edu/~keisler/calc.html}}.
* {{Citation | first=Anders|last= Kock|url=http://home.imf.au.dk/kock/sdg99.pdf|title= Synthetic Differential Geometry|publisher= Cambridge University Press|edition= 2nd|year=2006}}.
* {{Citation | author-link=William Lawvere|first=F.W.|last=Lawvere|url=http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf|title=Outline of synthetic differential geometry|publication-date=1998|year=1968}}.
* {{Citation | last1=Moerdijk|first1= I.|author1-link=Ieke Moerdijk|last2=Reyes|first2=G.E.|title=Models for Smooth Infinitesimal Analysis|publisher= Springer-Verlag|year= 1991}}.
* {{Citation | last1=Robinson | first1=Abraham | author1-link=Abraham Robinson | title=Non-standard analysis | publisher=[[Princeton University Press]] | isbn=978-0-691-04490-3 | year=1996}}.

== 関連項目 ==
{{Portal|数学}}
* {{仮リンク|無限小解析|en|Infinitesimal calculus}}
* [[微分方程式]]
* [[微分形式]]
* [[函数の微分]]
{{Infinitesimals}}

== 外部リンク ==
* {{MathWorld|urlname=Differential|title=Differential}}
* {{nlab|urlname=differential|title=differential}}
* {{PlanetMath|urlname=Differential|title=differential}}
* {{ProofWiki|urlname=Definition:Differential|title=Definition:Differential}}
* {{SpringerEOM|urlname=Differential|title=Differential}}

{{DEFAULTSORT:ひふんしよう}}
[[Category:初等解析学]]
[[Category:無限小解析]]
[[Category:数学に関する記事]]