# Definition of Dirichlet energy

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:

$$E[u]:=\frac{1}{2} \int_\Omega trace(du^t\cdot du)$$

• $$||\nabla u||^2=\nabla u\cdot\nabla u$$ Feb 18, 2019 at 19:54
• @MarkViola I'm only familiar with $\nabla u$ as gradient of function $u$ from $\mathbb R^n$ to $\mathbb R$, but according to definition above $u$ can be vector valued, in that case $\nabla u$ does not make sense to me. Is it some sort of generalisation? Thanks! Feb 18, 2019 at 20:26
• It might be a typo in the Spanish Wikipedia page you linked, since in the English page it has $u:\Omega \to \mathbb{R}$. Feb 18, 2019 at 20:37
• $\nabla u$ is a vector. Feb 18, 2019 at 20:41
• @angryavian, I thought so aswell, but in other notes it is defined is similiar way, see Def 2.4 page 5 Feb 18, 2019 at 20:43