- Weak derivative
mathematics, a weak derivative is a generalization of the concept of the derivativeof a function ("strong derivative") for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space. See distributions for an even more general definition.
Let be a function in the Lebesgue space . We say that in is a "weak derivative" of if,
for all continuously
differentiable functions with .
Generalizing to dimensions, if and are in the space of
locally integrable functions for some open set, and if is a multiindex, we say that is the -weak derivative of if
for all , that is, for all infinitely
differentiablefunctions with compact supportin . If has a weak derivative, it is often written since weak derivatives are unique (at least, up to a set of measure zero, see below).
absolute valuefunction "u" : [−1, 1] → [0, 1] , "u"("t") = |"t"|, which is not differentiable at "t" = 0, has a weak derivative "v" known as the sign functiongiven by
This is not the only weak derivative for "u": any "w" that is equal to "v"
almost everywhereis 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 measurezero, 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 equationsand in functional analysis.
*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.