please help me. I am learning the stability of flow by Navier - Stokes equations, but i have to show that for next exams
$\langle \vert \nabla\textbf{U}\vert ^2\rangle \equiv \langle \vert \nabla\textbf{U} : \nabla\textbf{U}\vert \rangle =\langle \vert curl\textbf{U}\vert ^2 \rangle = 2\langle \vert \textbf{D}:\textbf{D} \vert \rangle $
where
$\langle . \rangle = [\mathcal{M}(\mathcal{V})]^{-1}\int_{\mathcal{V}}(.)$ - $\mathcal{M}(\mathcal{V})$ is the contant measure of the volume $\mathcal{V}$.
$(\textbf{D})_{ij}=\dfrac{1}{2}(\partial_i U_j + \partial_jU_i)$ for all vector fields $\textbf{U} = (U_1(x,y,z);U_2(x,y,z);U_3(x,y,z))$ such that $div\textbf{U}= 0$ and $\textbf{U}\vert_{S}=0$ -- the boundary $S$ of a closed container $\mathcal{V}$
I'll very happy if you explain it to me. Thank you so much.