|
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 . See distributions for an even more general definition.
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 differentiable functions with compact support in . If has a weak derivative, it is often written since weak derivates are unique (at least, up to a set of measure zero, see below). Examples The function , which is not differentiable at t=0, has a weak derivative v known as the sign function given by ight. Properties It can be shown that 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 give 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 solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis. | ||||||||
|
| |||||||||
 |
|
| |