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Concept
Weak derivative
Summary
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 Lp space L^1([a,b]).
The method of integration by parts holds that for differentiable functions u and \varphi we have
:\begin{align}
\int_a^b u(x) \varphi'(x) , dx
& = \Big[u(x) \varphi(x)\Big]_a^b - \int_a^b u'(x) \varphi(x) , dx. \[6pt]
\end{align}
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 (\varphi(a)=\varphi(b)=0).
Definition
Let u be a function in the Lebesgue space L^1([a,b]). We say that v in L^1([a,b]) is a weak derivative of u if
:\int_a^b
Official source
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