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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 in 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
for all infinitely differentiable functions \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) of locally integrable functions for some open set 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 with compact support in 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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