I have recently come across the notion of weak directional derivatives in the context of Sobolev functions. Let $u\in W^{1,p}(\Omega)$ denote a Sobolev function for arbitrary exponent $p\geq 1$ and a domain $\Omega\subset\mathbb{R}^n$. Further, let $\nabla u$ denote its weak gradient. Is the definition of a weak directional derivative of $u$ in direction $\nu\in\mathbb{R}^n$ then simply $\partial_\nu u := \nabla u \cdot \nu$, which is a.e. defined? I can't think of anything else that would make sense right now. Maybe someone has a reference article/book concerning this topic.
Thanks in advance!