Differentiation Formula for Moving Regions. I've run into a few calculations in a series of textbooks/papers that require differentiating an integral with a changing region. In particular, I'd like to know if $f(x,t):\mathbb{R}^d\times \mathbb{R}\to \mathbb{R}$ is smooth (say), is 
$\frac{d}{dt}\int_{\{x: |{x}|\le t\}} f(x,t)\ dx=\int_{\{|x|=t\}}f(x,t)\cdot \frac{x}{|x|}dS(x)+\int_{\{|x|\le t\}}\frac{d}{dt}f(t,x)\ dx$?
Where $dS$ is surface measure of the ball. I realize this might be a particular instance of the Reynold transport formula, but I have yet to find a reference that gives an example of how to compute the velocity of the moving region. (Cf. Evans PDE Page 713)
 A: The differentiation formula for moving regions just enables you to interchange the  derivative and the integral. This can have important implications, as in the derivation of the governing partial differential equations of fluid dynamics.  
The velocity on the boundary is either (1) determined with a more extensive system of governing equations (eg., the velocity field in fluid motion) or (2) exogenously specified.
(1) In deriving the PDEs governing fluid motion, we invoke conservation of mass and momentum in a region where each point on the surface moves with the local fluid velocity $\mathbf{u}(\mathbf{x},t)$.  As you mention, this involves the Reynolds transport theorem.  If $\Omega(t) \subset   \mathbb{R}^3$ is the region, $\partial \Omega(t)$ is the boundary, and $\rho(\mathbf{x},t)$ is the fluid density, then 
$$\frac{d}{dt}\int_{\Omega(t)} \rho(\mathbf{x},t)d\mathbf{x}= \int_{\Omega(t)} \frac{\partial}{\partial t}\rho(\mathbf{x},t)d\mathbf{x}+ \int_{\partial\Omega(t)} \rho(\mathbf{x},t)\mathbf{u}(\mathbf{x},t)\cdot\mathbf{n}dS.$$
Using this equation along with the divergence theorem, we can derive the continuity equation
$$ \frac{\partial\rho}{\partial t}+ \nabla\cdot(\rho\mathbf{u})=0.$$
Additionally, the Navier-Stokes equations (expressing conservation of momentum) and boundary conditions are needed to close the system and actually solve for the velocity field.
(2) As a simple example of a specified surface velocity, consider a spherical balloon that expands as it is  filled with air at a constant volumetric rate $Q$.  The radius of the sphere $r(t)$ at time $t$ satisfies the differential equation
$$\frac{dr}{dt} = \frac{Q}{4\pi r^2}$$
Representing the balloon as a sphere centered at the origin in $\mathbb{R^3}$, the velocity at any point on the surface of this sphere has a velocity
$$\mathbf{u}(\mathbf{x},t)=\frac{Q}{4\pi [r(t)]^2}\mathbf{e}_r,$$
where the radial basis vector $\mathbf{e}_r$ also happens to be the outward normal vector to the surface.
