What is $\|\nabla u\|_{L^p}$?

Question

Let $u\in L(\mathbb R^d)$. Then, $\|u\|_{L^p}$ is $\left(\int_{\mathbb R^d}|u|^p\right)^{1/p}$. But since $\nabla u$ is a vector, what could be $\|\nabla u\|_{L^p}$ ? is it $$\int_{\mathbb R^d}(\nabla u\cdot \nabla u)^p=\int_{\mathbb R^d}\left(\sum_{i=1}^d (\partial_i u)^2\right)^p\ \ ?$$ Honestely, it looks wrong...