Real functions Functions of several real variables Concept

Total derivative

Summary

Let $f$ map an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ . The total derivative of $f$ at a point $a$ is the linear map $Df(a) \colon \mathbb{R}^n \to \mathbb{R}^m$ that best approximates $f$ near $a$ . That is, $f(a + h) = f(a) + Df(a)(h) + o(\|h\|)$ as $h \to 0$ . It is the Fréchet derivative in the case of Euclidean spaces. Its matrix in the standard bases is the Jacobian matrix of $f$ at $a$ .

Remarks

The total derivative accounts for changes in all variables at once. A partial derivative accounts for a change in one variable only. When $f$ depends on $t$ both directly and through other functions of $t$ , the total derivative $\dfrac{d}{dt} f(t, x(t))$ is the sum $\dfrac{\partial f}{\partial t} + \dfrac{\partial f}{\partial x} \dfrac{dx}{dt}$ , by the chain rule.

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Euclidean space object
Total derivative concept

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Tag	`0FNKFT8H6E`
URI	`m2://0FNKFT8H6E`
Aliases	`derivative-of-a-map-between-euclidean-spaces`, `differential-of-a-map`
Kind	concept
Writing	W1 stated
MSC	26B05
Uses	Euclidean space
Same as	Wikidata Q636889 Wikipedia Derivative (multivariable calculus)

Total derivative

Summary

Remarks

Reading path

Where this leads

Further reading

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