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Weak derivative

 

 
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Weak derivative
 
 
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In mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, a weak derivative is a generalization of the concept of the derivativeDerivative

In mathematics, the derivative is defined as the instantaneous rate of change of a function....
 of a functionFunction (mathematics)

In mathematics, a function relates each of its inputs to exactly one output....
 (strong derivative) for functions not assumed differentiable, but only integrableIntegrable function

In mathematics, the term integrable function refers to a function whose integral exists....
, i.e. to lie in the Lebesgue space . See distributionDistribution (mathematics)

In mathematical analysis, distributions are objects which generalize functions and probability distributions....
s 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 functionLocally integrable function

In mathematics, a locally integrable function is a function which is integrable on any compact set....
s for some open setOpen set

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "cha...
 , 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 derivatives are unique (at least, up to a set of measure zeroMeasure zero

Let μ be a measure on a sigma algebra Σ of subsets of a set X....
, see below).

Examples

The absolute valueAbsolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its sign....
 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 functionSign function

In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real...
 given by

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

In measure theory , one says that a property holds almost everywhere if the set of elements for which the property does no...
 is also a weak derivative for u. Usually, this is not a problem, since in the theory of Lp spacesLp space

In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding seque...
 and Sobolev spaceSobolev space

In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivat...
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 measureLebesgue measure

In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space...
 zero, i.e., they are equal almost everywhereAlmost everywhere

In measure theory , one says that a property holds almost everywhere if the set of elements for which the property does no...
. 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 solutionWeak solution

In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives appe...
s in Sobolev spaceSobolev space

In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivat...
s, which are useful for problems of differential equations and in functional analysisFunctional analysis

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functi...
.

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