PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very low Entry average rating: No information on entry rating
weak derivative (Definition)

Let $ f\colon \Omega\to \mathbf R$ and $ g=(g_1,\ldots,g_n)\colon \Omega\to \mathbf R^n$ be locally integrable functions defined on an open set $ \Omega\subset \mathbf R^n$. We say that $ g$ is the weak derivative of $ f$ if the equality

$\displaystyle \int_\Omega f \frac{\partial \phi}{\partial x_i} = - \int_\Omega g_i \phi $
holds true for all functions $ \phi\in\mathcal C^\infty_c(\Omega)$ (smooth functions with compact support in $ \Omega$) and for all $ i=1,\ldots, n$. Notice that the integrals involved are well defined since $ \phi$ is bounded with compact support and because $ f$ is assumed to be integrable on compact subsets of $ \Omega$.

Comments

  1. If $ f$ is of class $ \mathcal C^1$ then the gradient of $ f$ is the weak derivative of $ f$ in view of Gauss Green Theorem. So the weak derivative is an extension of the classical derivative.
  2. The weak derivative is unique (as an element of the Lebesgue space $ L^1_{\mathrm loc}$) in view of a result about locally integrable functions.
  3. The same definition can be given for functions with complex values.



"weak derivative" is owned by paolini.
(view preamble)

View style:

See Also: Sobolev space
Log in to rate this entry.
(view current ratings)

Cross-references: complex, derivative, extension, Gauss Green theorem, gradient, class, compact subsets, support, compact, bounded, well defined, integrals, smooth functions with compact support, functions, equality, open set, locally integrable functions
There are 4 references to this entry.

This is version 13 of weak derivative, born on 2004-12-27, modified 2007-06-29.
Object id is 6600, canonical name is WeakDerivative.
Accessed 4173 times total.

Classification:
AMS MSC: 46E35 (Functional analysis :: Linear function spaces and their duals :: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems)
Pending Errata and Addenda
None.
[ View all 6 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)