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In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e., to lie in the L space L 1 ] {\displaystyle L^{1}]}.
The method of integration by parts holds that for differentiable functions u {\displaystyle u} and φ {\displaystyle \varphi } 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 = φ = 0 {\displaystyle \varphi =\varphi =0} ].
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