I don't understand definition of Dirichlet energy
$$ E[u]:=\frac{1}{2} \int_\Omega \Vert \nabla u \Vert^2 $$
for $u:\Omega \to \mathbb R^n$ with $\Omega \subset \mathbb R^n$.
Let's consider for example $u: \Omega \to \mathbb R^2$. What would
$$ \Vert \nabla u \Vert^2 $$ mean? Can we say it's determinant of Jacobi Matrix of $u$ or rather quadratic sum of all components of it?
Edit: for the people wondering Dirichlet energy in general is defined as:
$$E[u]:=\frac{1}{2} \int_\Omega trace(du^t\cdot du)$$