weak derivative
Let $f:\mathrm{\Omega}\to \mathbf{R}$ and $g=({g}_{1},\mathrm{\dots},{g}_{n}):\mathrm{\Omega}\to {\mathbf{R}}^{n}$ be locally integrable functions defined on an open set $\mathrm{\Omega}\subset {\mathbf{R}}^{n}$. We say that $g$ is the weak derivative of $f$ if the equality
$${\int}_{\mathrm{\Omega}}f\frac{\partial \varphi}{\partial {x}_{i}}={\int}_{\mathrm{\Omega}}{g}_{i}\varphi $$ 
holds true for all functions $\varphi \in {\mathcal{C}}_{c}^{\mathrm{\infty}}(\mathrm{\Omega})$ (smooth functions with compact support in $\mathrm{\Omega}$) and for all $i=1,\mathrm{\dots},n$. Notice that the integrals involved are well defined since $\varphi $ is bounded with compact support and because $f$ is assumed to be integrable on compact subsets of $\mathrm{\Omega}$.
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 ${L}_{\mathrm{l}oc}^{1}$) in view of a result about locally integrable functions.

3.
The same definition can be given for functions with complex values.
Title  weak derivative 

Canonical name  WeakDerivative 
Date of creation  20130322 14:54:52 
Last modified on  20130322 14:54:52 
Owner  paolini (1187) 
Last modified by  paolini (1187) 
Numerical id  16 
Author  paolini (1187) 
Entry type  Definition 
Classification  msc 46E35 
Related topic  SobolevSpaces 