# Difference between revisions of "Generalized derivative"

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− | Another equivalent definition of the generalized derivative $\frac{\partial f}{\partial x_j}$ is the following. If | + | Another equivalent definition of the generalized derivative $\frac{\partial f}{\partial x_j}$ is the following. If $f$ can be modified on a set of $n$-dimensional measure zero so that the modified function (which will again be denoted by $f$) is locally [[Absolutely_continuous_function#Absolute_continuity_of_a_function|absolutely continuous]] with respect to $x_j$ for almost-all (in the sense of the $(n-1)$-dimensional Lebesgue measure) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379021.png" /> belonging to the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379023.png" /> onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379025.png" /> has partial derivative (in the usual sense of the word) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379026.png" /> [[Almost-everywhere|almost-everywhere]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379027.png" />. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379028.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379030.png" /> is a generalized derivative of <img align="absmi |

ddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379031.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379033.png" />. Thus, a generalized derivative is defined almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379034.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379035.png" /> is continuous and the ordinary derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379036.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379037.png" />, then it is also a generalized derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379038.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379040.png" />. | ddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379031.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379033.png" />. Thus, a generalized derivative is defined almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379034.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379035.png" /> is continuous and the ordinary derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379036.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379037.png" />, then it is also a generalized derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379038.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043790/g04379040.png" />. | ||

## Revision as of 14:26, 21 November 2012

*of function type*

An extension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev (see [1], [2]), who arrived at a definition of a generalized derivative from the point of view of his concept of a generalized function.

Let $f$ and $\phi$ be locally integrable functions on an open set $\Omega\subset \mathbb R^n$, that is, Lebesgue integrable on any closed bounded set $F\subset\Omega$. Then $\phi$ is the generalized derivative of $f$ with respect to $x_j$ on $\Omega$, and one writes $\phi = \frac{\partial f}{\partial x_j}$, if for any infinitely-differentiable function $\psi$ with compact support in $\Omega$ (see Function of compact support)

\begin{equation}\label{eq:1} \int\limits_{\Omega}f(x)\frac{\partial \psi}{\partial x_j}(x)\,dx = -\int\limits_{\Omega}\phi(x) \psi(x)\,dx. \end{equation}

Another equivalent definition of the generalized derivative $\frac{\partial f}{\partial x_j}$ is the following. If $f$ can be modified on a set of $n$-dimensional measure zero so that the modified function (which will again be denoted by $f$) is locally absolutely continuous with respect to $x_j$ for almost-all (in the sense of the $(n-1)$-dimensional Lebesgue measure) belonging to the projection of onto the plane , then has partial derivative (in the usual sense of the word) almost-everywhere on . If a function almost-everywhere on , then is a generalized derivative of with respect to on . Thus, a generalized derivative is defined almost-everywhere on ; if is continuous and the ordinary derivative is continuous on , then it is also a generalized derivative of with respect to on .

Generalized derivatives of a higher order are defined by induction. They are independent (almost-everywhere) of the order of differentiation.

There is a third equivalent definition of a generalized derivative. Suppose that for each closed bounded set , the functions and , defined on , have the properties:

and suppose that the functions , and their partial derivatives are continuous on . Then is the generalized partial derivative of with respect to on () (see also Sobolev space).

From the point of view of the theory of generalized functions, a generalized derivative can be defined as follows: Suppose one is given a function that is locally summable on , considered as a generalized function, and let be the partial derivative in the sense of the theory of generalized functions. If represents a function that is locally summable on , then is a generalized derivative (in the first (original) sense).

The concept of a generalized derivative had been considered even earlier (see [3] for example, where generalized derivatives with integrable square on are considered). Subsequently, many investigators arrived at this concept independently of their predecessors (on this question see [4]).

#### References

[1] | S.L. Sobolev, "Le problème de Cauchy dans l'espace des fonctionnelles" Dokl. Akad. Nauk SSSR , 3 : 7 (1935) pp. 291–294 |

[2] | S.L. Sobolev, "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales" Mat. Sb. , 1 (1936) pp. 39–72 |

[3] | B. Levi, "Sul principio di Dirichlet" Rend. Circ. Mat. Palermo , 22 (1906) pp. 293–359 Zbl 37.0414.06 Zbl 37.0414.04 |

[4] | S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) Zbl 0307.46024 |

#### Comments

#### References

[a1] | S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965) MR0178246 Zbl 0142.37401 |

**How to Cite This Entry:**

Generalized derivative.

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