Weak derivative

Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function ("strong derivative") for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1( [a,b] ). See distributions for an even more general 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'(t)dt=-int_a^b v(t)varphi(t)dt

for all continuously differentiable functions 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 functions for some open set U subset mathbb{R}^n, and if alpha is a multiindex, 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 with compact 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 of measure zero, see below).


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 the sign function given by

:v colon [-1,1] o [-1,1] colon t mapsto v(t) = egin{cases} 1, & mbox{if } t > 0; \ 0, & mbox{if } t = 0; \ -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 and Sobolev spaces, functions that are equal almost everywhere are identified.


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 equal almost 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.


This concept gives rise to the definition weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.

ee also



*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

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Weak formulation — Weak formulations are an important tool for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation …   Wikipedia

  • Derivative — This article is an overview of the term as used in calculus. For a less technical overview of the subject, see Differential calculus. For other uses, see Derivative (disambiguation) …   Wikipedia

  • Weak solution — In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives appearing in the equation may not all exist but which is nonetheless deemed to satisfy… …   Wikipedia

  • Derivative of a constant — In calculus, the derivative of a constant function is zero (A constant function is one that does not depend on the independent variable, such as f(x) = 7). The rule can be justified in various ways. The derivative is the slope of the tangent to… …   Wikipedia

  • weak — [13] Etymologically, something that is weak is ‘bendable’. The word was borrowed from Old Norse veikr. This was descended from prehistoric Germanic *waikwaz, which also produced German weich and Dutch week ‘soft’. And this in turn was formed from …   The Hutchinson dictionary of word origins

  • weak — [13] Etymologically, something that is weak is ‘bendable’. The word was borrowed from Old Norse veikr. This was descended from prehistoric Germanic *waikwaz, which also produced German weich and Dutch week ‘soft’. And this in turn was formed from …   Word origins

  • Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… …   Wikipedia

  • Bounded variation — In mathematical analysis, a function of bounded variation refers to a real valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a… …   Wikipedia

  • Sobolev space — In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense… …   Wikipedia

  • List of mathematics articles (W) — NOTOC Wad Wadge hierarchy Wagstaff prime Wald test Wald Wolfowitz runs test Wald s equation Waldhausen category Wall Sun Sun prime Wallenius noncentral hypergeometric distribution Wallis product Wallman compactification Wallpaper group Walrasian… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”