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weak derivative ：ウィキペディア英語版

weak derivative

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 L^''p'' space

\mathrm^1(())

. See distributions for an even more general definition.
== Definition ==

Let

u

be a function in the Lebesgue space

L^1(())

. We say that

v

L^1(())

is a ''weak derivative'' of

u

if,
:

\int_a^b u(t)\varphi'(t)dt=-\int_a^b v(t)\varphi(t)dt

\varphi

with

\varphi(a)=\varphi(b)=0

. This definition is motivated by the integration technique of Integration by parts.
Generalizing to

n

dimensions, if

u

and

v

are in the space

L_^1(U)

U \subset \mathbb^n

, and if

\alpha

is a multi-index, we say that

v

is the

\alpha^

-weak derivative of

u

if
:

\int_U u D^ \varphi=(-1)^ \int_U v\varphi

for all

\varphi \in C^_c (U)

, that is, for all infinitely differentiable functions

\varphi

U

. If

u

has a weak derivative, it is often written

D^u

since weak derivatives are unique (at least, up to a set of measure zero, see below).

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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