Why does the material derivative and transport theorem look different? Reynolds transport theorem says that 
$
\frac{d\int\phi}{dt}=\int\left(\frac{\partial\phi}{\partial t} + \nabla\cdot(\phi\otimes v) \right)
$
Why is the material derivative not defined as what's inside the integral on the right hand side?
 A: The Reynolds transport theorem reads
$$\frac{\mathrm{d}}{\mathrm{d} t} \int_{\Omega} \phi\, \mathrm{d}V= \int_{\Omega} \frac{\partial \phi}{\partial t} \, \mathrm{d}V + \int_{\partial \Omega} \phi \left[ \mathbf{v} \cdot \mathbf{n} \right] \mathrm{d}A$$
The divergence theorem then says
$$ \int_{\partial \Omega} \phi \left[ \mathbf{v} \cdot \mathbf{n} \right] \mathrm{d}A = \int_{\Omega} \mathrm{div} \left( \phi \mathbf{v} \right)  \mathrm{d}V $$
Moreover,
$$ \mathrm{div}\left( \phi \mathbf{v} \right) = \mathrm{grad}\left( \phi \right) \cdot \mathbf{v} + \phi \, \mathrm{div}\left(\mathbf{v} \right) $$
The term $\phi \, \mathrm{div}\left(\mathbf{v} \right)$ is a source or sink of $\phi$.
Compare the above with the material derivative
$$ \frac{\mathrm{D} \phi}{\mathrm{D} t} = \frac{\partial \phi}{\partial t} + \mathrm{grad}\left( \phi \right) \cdot \mathbf{v} $$
It describes the time rate of change of $\phi$. For a conservation law, this change is balanced by the sum of all sources and sinks of $\phi$. The conservation of mass in fluid mechanics is a nice example:
$$\frac{\partial \rho}{\partial t} + \mathrm{div}\left( \rho \mathbf{v} \right) = \frac{\partial \rho}{\partial t} + \mathrm{grad}\left( \rho \right) \cdot \mathbf{v} + \rho \, \mathrm{div}\left(\mathbf{v} \right) = 0 $$
The time rate of change of mass density $\rho$ is given by $\partial \rho / \partial t + \mathrm{grad}\left( \rho \right) \cdot \mathbf{v}$, which is balanced by the source/sink $\rho \, \mathrm{div}\left(\mathbf{v} \right)$.
The Reynolds transport theorem and the divergence theorem describe the conservation of a physical quantity that can be transported by a flux. The source/sink of that quantity, that is created by the flux velocity, is already included in this description. The material derivative instead gives the time rate of change of any quantity, even those that are not conserved, for example temperature.
