Q$1.$ For a scalar field $=(x,t)$ show that the following relationship holds:

$\frac{d}{dt}{\displaystyle \underset{S_t}{\overset{\phantom{}}{}}}nda={\displaystyle \underset{S_t}{\overset{\phantom{}}{}}}(\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}+$div$(v)-L^T)nda,$

where $v$ is the velocity and $L$ is the velocity gradient.