Definition
Let be a function in the Lebesgue space . We say that in is a weak derivative of if,
for all infinitely differentiable functions with . This definition is motivated by the integration technique of Integration by parts.
Generalizing to dimensions, if and are in the space of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
for all, that is, for all infinitely differentiable functions with compact support in . If has a weak derivative, it is often written since weak derivatives are unique (at least, up to a set of measure zero, see below).
Read more about this topic: Weak Derivative
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