Is it okay to write like this in integral?

Question

$u,w\in W_{0}^{1,p}(\Omega)$, where $\Omega$ is a bounded smooth domain. I wrote like this: $$\int_{\Omega} |\nabla u(x)|^{p-2} \Big| \sum_{i=1}^{n} \partial_i u(x) \partial_i w(x) \Big| dx =\int_{\Omega} |\nabla u(x)|^{p-2} |\nabla u(x) \cdot \nabla w(x)| dx,$$ where $a \cdot b$ denotes the inner product of $a,b\in\mathbb{R}^n$. Does this way of writing make any sense?