How can I evaluate the material derivative of $\frac{\|\vec{v}\|^2}{2}$ ?
This question arises from the transport of the kinematic energy of a fluid:
$$\frac{d}{dt}\int_{B_t} \rho \frac{\|\vec{v}\|^2}{2}\ dV = \int_{B_t} \rho \dot{\left( \frac{\|\vec{v}\|^2 }{2}\right)} \ dV$$
I know that the material derivative acts as $\partial_t (-) + \vec{v}\cdot\vec{\nabla}(-)$ but I'm not able to evaluate it over a norm.
