I need to compute the norm of the function $f : \mathbb R^2 \to \mathbb R^2$ in the Sobolev space $H^1$.
Could somebody tell me if the following definition is correct? For example, for function $f$ given by $(x, y) \to (u, v)$
$$ \| f(x, y) \|_{H^1}^2 = \int |u|^2 + \int |v|^2 + \int |\partial_x u|^2 + \int |\partial_y u |^2 + \int |\partial_x v|^2 + \int |\partial_y v|^2. $$
When I look at the textbooks, they always operate on functionals, that is, functions with range in $\mathbb R$.