Second Problem

Show that the derivative of an integral is given by

$$\frac{d}{d t} \int\limits_{x_1(t)}^{x_2(t)}\phi(x,t) d x = \int\l... ...{\partial t} d x + \frac{d x_2}{d t}\phi(x_2,t) - \frac{d x_1}{d t}\phi(x_1,t).$$ (1)

By generalizing to three dimensions show that the material derivative of an integral of fluid property is given by

$$\frac{D}{D t} \int_V \phi(x,t) d V = \int \frac{\partial \phi}{\p... ...t_V\left[\frac{\partial \phi}{\partial t} + \nabla\cdot({\bf v}\phi)\right]d V.$$ (2)

where the surface integral $\int_S$ is over the surface bounding the volume $V$. Hence show that

$$\frac{D}{D t} \int_V \rho \phi d V = \int_V \rho \frac{D\phi}{D t} d V.$$

Hint You may find it useful to Google Leibnitz's rule and Reynolds transport theorem.