- Weak derivative
In

mathematics , a**weak derivative**is a generalization of the concept of thederivative of a function ("strong derivative") for functions not assumeddifferentiable , but only integrable, i.e. to lie in theLebesgue space $L^1(\; [a,b]\; )$. See distributions for an even more general definition.**Definition**Let $u$ be a function in the Lebesgue space $L^1(\; [a,b]\; )$. We say that $v$ in $L^1(\; [a,b]\; )$ is a "weak derivative" of $u$ if,

:$int\_a^b\; u(t)varphi\text{'}(t)dt=-int\_a^b\; v(t)varphi(t)dt$

for

**all**continuouslydifferentiable function s $varphi$ with $varphi(a)=varphi(b)=0$.Generalizing to $n$ dimensions, if $u$ and $v$ are in the space $L\_\{loc\}^1(U)$ of

locally integrable function s for someopen set $U\; subset\; mathbb\{R\}^n$, and if $alpha$ is amultiindex , we say that $v$ is the $alpha^\{th\}$-weak derivative of $u$ if:$int\_U\; u\; D^\{alpha\}\; varphi=(-1)^\; int\_U\; vvarphi$

for all $varphi\; in\; C^\{infty\}\_c\; (U)$, that is, for all infinitely

differentiable functions $varphi$ withcompact support in $U$. If $u$ has a weak derivative, it is often written $D^\{alpha\}u$ since weak derivatives are unique (at least, up to a set ofmeasure zero , see below).**Examples**The

absolute value function "u" : [−1, 1] → [0, 1] , "u"("t") = |"t"|, which is not differentiable at "t" = 0, has a weak derivative "v" known as thesign function given by:$v\; colon\; [-1,1]\; o\; [-1,1]\; colon\; t\; mapsto\; v(t)\; =\; egin\{cases\}\; 1,\; mbox\{if\; \}\; t\; 0;\; \backslash \; 0,\; mbox\{if\; \}\; t\; =\; 0;\; \backslash \; -1,\; mbox\{if\; \}\; t\; 0.\; end\{cases\}$

This is not the only weak derivative for "u": any "w" that is equal to "v"

almost everywhere is also a weak derivative for "u". Usually, this is not a problem, since in the theory of "L"^{"p"}spaces andSobolev space s, functions that are equal almost everywhere are identified.**Properties**If two functions are weak derivatives of the same function, they are equal except on a set with

Lebesgue measure zero, i.e., they are equalalmost everywhere . If we consider equivalence classes of functions, where two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique.Also, if "u" is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative.

**Extensions**This concept gives rise to the definition

weak solution s inSobolev space s, which are useful for problems ofdifferential equations and infunctional analysis .**ee also***

subderivative **References***

*Cite book | author=Evans, Lawrence C. | authorlink= | coauthors= | title=Partial differential equations | date=1998 | publisher=American Mathematical Society | location=Providence, R.I. | isbn=0-8218-0772-2 | pages=page 242

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