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 L¹([a,b]). Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the derivative is one of the two central concepts of calculus. ... Partial plot of a function f. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Integrability is a mathematical concept used in different areas. ... The title given to this article is incorrect due to technical limitations. ...

Definition

Let u be a function in the Lebesgue space L¹([a,b]). We say that v in L¹([a,b]) is a weak derivative of u if,

for all continuously differentiable functions with .

Generalizing to n dimensions, if u and v are in the space of locally integrable functions for some open set , and if α is a multiindex, we say that v is the α^th-weak derrivative of u if In mathematics, a locally integrable function is a function which is integrable on any compact set. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... The notion of multi-indices simplifies formulae used in the multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an array of indices. ...

for all , that is, for all infinitely differentiable functions with compact support in U. If u has a weak derivative, it is often written D^αu since weak derivates are unique (at least, up to a set of measure zero, see below). In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ...

Examples

The function , which is not differentiable at t=0, has a weak derivative v known as the sign function given by :

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 functions that are equal almost everywhere to be identical, we can say that the weak derivative is unique. In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

Another important fact is that if a function u has a weak derivative then u is continuous. 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.

The fundamental theorem for weak derivatives states that the function is the weak derivative of if and only if

.

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. 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 the equation in some precisely defined sense. ... In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...