In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a weak derivative is a generalization of the concept of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
(''strong derivative'') for functions not assumed
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, but only
integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, i.e., to lie in the
L''p'' space .
The method of
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
holds that for differentiable functions
and
we have
:
A function ''u''
' being the weak derivative of ''u'' is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions ''φ'' vanishing at the boundary points (
).
Definition
Let
be a function in the
Lebesgue space . We say that
in
is a weak derivative of
if
:
for ''all'' infinitely
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
s
with
.
Generalizing to
dimensions, if
and
are in the space
of
locally integrable function In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies ...
s for some
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
, and if
is a
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
, we say that
is the
-weak derivative of
if
:
for all
, that is, for all infinitely differentiable functions
with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
in
. Here
is defined as
If
has a weak derivative, it is often written
since weak derivatives are unique (at least, up to a set of
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
, see below).
Examples
*The
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function
, which is not differentiable at
has a weak derivative
known as the
sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
, and given by
This is not the only weak derivative for ''u'': any ''w'' that is equal to ''v''
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
is also a weak derivative for ''u''. (In particular, the definition of ''v''(0) above is superfluous and can be replaced with any desired real number r.) Usually, this is not a problem, since in the theory of
''L''''p'' spaces and
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s, functions that are equal almost everywhere are identified.
*The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of the rational numbers
is nowhere differentiable yet has a weak derivative. Since the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
of the rational numbers is zero,
Thus
is a weak derivative of
. Note that this does agree with our intuition since when considered as a member of an Lp space,
is identified with the zero function.
*The
Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Th ...
''c'' does not have a weak derivative, despite being differentiable almost everywhere. This is because any weak derivative of ''c'' would have to be equal almost everywhere to the classical derivative of ''c'', which is zero almost everywhere. But the zero function is not a weak derivative of ''c'', as can be seen by comparing against an appropriate test function
. More theoretically, ''c'' does not have a weak derivative because its
distributional derivative
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
, namely the
Cantor distribution
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.
This distribution has neither a probability density function nor a probability mass function, since although its cumulativ ...
, is a
singular measure In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on ...
and therefore cannot be represented by a function.
Properties
If two functions are weak derivatives of the same function, they are equal except on a set with
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
zero, i.e., they are equal
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. If we consider
equivalence classes
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of functions such that 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 of
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 may not all exist but which is nonetheless deemed to satisfy the equation in some precis ...
s in
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s, which are useful for problems of
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
and in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
See also
*
Subderivative
In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection ...
*
Weyl's lemma (Laplace equation)
References
*
*
* {{Cite book , author1=Knabner, Peter , author2=Angermann, Lutz , title=Numerical methods for elliptic and parabolic partial differential equations , url=https://archive.org/details/numericalmethods00knab , url-access=limited , year=2003 , publisher=Springer , location=New York , isbn=0-387-95449-X , pag
53}
Generalized functions
Functional analysis
Generalizations of the derivative
Generalizations