Editing Differential (mathematics) (section)

==== 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.