Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts coordinate wise, and we integrate by parts, I wish to write this as $-\int_{\mathbb T^2} \nabla \mathbb v \cdot \nabla \mathbb v$. Is this the right notation for this? Or must I use a double dot for this? What is the right notation for this?
Thank you.