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The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. For example, if ''x'' is a variable, then a change in the value of ''x'' is often denoted Δ''x'' (pronounced ''delta x''). The differential d''x'' represents an infinitely small change in the variable ''x''. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If ''y'' is a function of ''x'', then the differential d''y'' of ''y'' is related to d''x'' by the formula : where d''y''/d''x'' 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 Δ''y''/Δ''x'' as Δ''x'' becomes infinitesimal. 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.〔See and .〕 # 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.〔See and .〕 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. == History and usage == 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.〔.〕 Isaac Newton referred to them as 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'' ′. 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 of the graph's tangent line), which may be obtained by taking the 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 integrals 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 : 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Differential (infinitesimal)」の詳細全文を読む スポンサード リンク
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