Differential (math)

Differential (from lat. Differentia “difference, difference”) is the linear part of the increment of the function .

Content

Conventions

Usually function differential $f$ ${\ displaystyle f}$ is indicated $d f$ ${\ displaystyle df}$ . Some authors prefer to label ${\rm {d}}f$ ${\ displaystyle {\ rm {d}} f}$ direct font, wanting to emphasize that the differential is an operator .

Differential at point $x_{0}$ ${\ displaystyle x_ {0}}$ is indicated $d_{x_{0}}f$ ${\ displaystyle d_ {x_ {0}} f}$ , and sometimes $df_{x_{0}}$ ${\ displaystyle df_ {x_ {0}}}$ or $df[x_{0}]$ ${\ displaystyle df [x_ {0}]}$ , and $d f$ ${\ displaystyle df}$ if value $x_{0}$ ${\ displaystyle x_ {0}}$ clear from the context.

Accordingly, the differential value at the point $x_{0}$ ${\ displaystyle x_ {0}}$ from $h$ ${\ displaystyle h}$ may be denoted as $d_{x_{0}}f(h)$ ${\ displaystyle d_ {x_ {0}} f (h)}$ , and sometimes $df_{x_{0}}(h)$ ${\ displaystyle df_ {x_ {0}} (h)}$ or $df[x_{0}](h)$ ${\ displaystyle df [x_ {0}] (h)}$ , and $df(h)$ ${\ displaystyle df (h)}$ if value $x_{0}$ ${\ displaystyle x_ {0}}$ clear from the context.

Using the differential sign

The differential sign is used in the expression for the integral $\int f(x)\,dx$ ${\ displaystyle \ int f (x) \, dx}$ . Moreover, sometimes (and not quite correctly) the differential $d x$ ${\ displaystyle dx}$ introduced as part of the definition of the integral .
The differential sign is also used in the Leibniz notation for the derivative $f'(x_{0})={\frac {df}{dx}}(x_{0})$ ${\ displaystyle f '(x_ {0}) = {\ frac {df} {dx}} (x_ {0})}$ . This notation is motivated by the fact that for differentials, functions $f$ ${\ displaystyle f}$ and identical function $x$ ${\ displaystyle x}$ true ratio
$d_{x_{0}}f=f'(x_{0}){\cdot }d_{x_{0}}x.$ ${\ displaystyle d_ {x_ {0}} f = f '(x_ {0}) {\ cdot} d_ {x_ {0}} x.}$

Definitions

For functions

Function differential $f\colon \mathbb {R} \to \mathbb {R}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$ at the point $x_{0}\in \mathbb {R}$ ${\ displaystyle x_ {0} \ in \ mathbb {R}}$ can be defined as a linear function

d_{x_{0}}f(h)=f'(x_{0})h,

{\ displaystyle d_ {x_ {0}} f (h) = f '(x_ {0}) h,}

Where $f'(x_{0})$ ${\ displaystyle f '(x_ {0})}$ denotes a derivative $f$ ${\ displaystyle f}$ at the point $x_{0}$ ${\ displaystyle x_ {0}}$ , but $h$ ${\ displaystyle h}$ - increment of the argument when moving from $x_{0}$ ${\ displaystyle x_ {0}}$ to $x_{0}+h$ ${\ displaystyle x_ {0} + h}$ .

In this way $d f$ ${\ displaystyle df}$ there is a function of two arguments $df\colon (x_{0},h)\mapsto d_{x_{0}}f(h)$ ${\ displaystyle df \ colon (x_ {0}, h) \ mapsto d_ {x_ {0}} f (h)}$ .

The differential can be determined directly, that is, without involving the definition of a derivative, as a function $d_{x_{0}}f(h)$ ${\ displaystyle d_ {x_ {0}} f (h)}$ linearly dependent on $h$ ${\ displaystyle h}$ , and for which the following relation is true

d_{x_{0}}f(h)=f(x_{0}+h)-f(x_{0})+o(h).

{\ displaystyle d_ {x_ {0}} f (h) = f (x_ {0} + h) -f (x_ {0}) + o (h).}

For displays

Differential display $f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}$ ${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}}$ at the point $x_{0}\in \mathbb {R} ^{n}$ ${\ displaystyle x_ {0} \ in \ mathbb {R} ^ {n}}$ called a linear operator $d_{x_{0}}f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}$ ${\ displaystyle d_ {x_ {0}} f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}}$ such that the condition

d_{x_{0}}f(h)=f(x_{0}+h)-f(x_{0})+o(h).

{\ displaystyle d_ {x_ {0}} f (h) = f (x_ {0} + h) -f (x_ {0}) + o (h).}

Related Definitions

Display $f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}$ ${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}}$ called differentiable at the point $x_{0}\in \mathbb {R} ^{n}$ ${\ displaystyle x_ {0} \ in \ mathbb {R} ^ {n}}$ if differential defined $d_{x_{0}}f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}$ ${\ displaystyle d_ {x_ {0}} f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}}$ .

Properties

Linear operator matrix $d$ $x$ $0$ $f$ {\ displaystyle d_ {x_ {0}} f} equal to the Jacobi matrix ; its elements are partial derivatives $f$ {\ displaystyle f} .
- Note that the Jacobi matrix can be defined at the point where the differential is not defined.
Function differential $f$ ${\ displaystyle f}$ connected with its gradient $\nabla f$ ${\ displaystyle \ nabla f}$ the following defining relation
$d_{x_{0}}f(h)=\langle (\nabla f)(x_{0}),h\rangle$ ${\ displaystyle d_ {x_ {0}} f (h) = \ langle (\ nabla f) (x_ {0}), h \ rangle}$

History

The term "differential" is introduced by Leibniz . Originally $d x$ ${\ displaystyle dx}$ used to mean " infinitesimal " - a quantity that is less than any finite quantity and yet non-zero. This view was inconvenient in most branches of mathematics, with the exception of non-standard analysis .

Variations and generalizations

The concept of differential contains more than just the differential of a function or display. It can be generalized by obtaining various important objects in functional analysis, differential geometry, measure theory, non-standard analysis, algebraic geometry, and so on.

Differential (differential geometry)
Higher Order Differentials
Ito differential
External differential
Peano derivative
Frechet derivative

Literature

G. M. Fichtengolts “The course of differential and integral calculus”