How to compute the $L^{2}$ error of the gradient in the Finite Element Method

Let $Omegasubset mathbb{R}^{2}$ and $tau_{h} = {Omega_{k}}_{k=1}^{N}$ be a triangulation of $Omega$. The $L^2$ error for a FEM approximation $u_{h}$ is given by:

$ || u-u_{h} ||_{L^2} = sqrt{ int_{Omega} left( u-u_{h} right)^2 } = sqrt{ sum_{k=1}^{N} int_{Omega_{k}} left( u-u_{h}^{k} right)^2 } $.

Where $u_{h}^{k}$ is the approximation in the element $Omega_{k}$. Suppose that I also have an approximation $q_{h} = ( q_{1h}, q_{2h} )$ for the gradient $q = ( q_{1}, q_{2} ) = nabla u$.

My question is: how do you compute the $L^{2}$ error for the gradient? i.e, $|| q-q_{h} ||_{L^2} =$ ?

I was thinking in something like $|| q-q_{h} ||_{L^2} = sqrt{int_{Omega} left( q_{1}-q_{1h}^{k} right)^2+int_{Omega} left( q_{2}-q_{2h}^{k} right)^2 } $, but I’m not sure if it is correct. Thanks!