Flow term in Reynolds transport theorem The basic form of Reynolds transport theorem for a fixed control volume can be written as 
$$\frac{D B_{sys}}{Dt}=\frac{\partial B_{CV}}{\partial t}-\dot{B_{in}}+\dot{B_{out}}$$
Why is the $\dot{B_{in}}$ term negative and the $\dot{B_{out}}$ term positive instead of the other way around? Doesn't a flow of $B$ into the system increment that property? 
My textbook says "In the limit of time $\mathrm dt\to 0$, instantaneous change of $B$ in the system is the sum of change within, plus the outflow, minus the inflow." That would mean $\frac{D B_{sys}}{Dt}$ represents the rate of change of $B$ in the system as opposed to an external source term. The exact equation given there is
$$\frac{\mathrm d(B_{sys})}{\mathrm dt}= \frac{\mathrm d}{\mathrm dt} \left(\,\int\limits_{CV} \beta \rho \,\mathrm d \upsilon\right) + \int\limits_{CS} \beta \rho V \cos \theta \,\mathrm dA_{out} - \int\limits_{CS} \beta \rho V \cos \theta \,\mathrm dA_{in}$$
Is the textbook wrong then?
 A: It's just a matter of convention. The in and out terms are based on the dot product of the velocity vector and the surface normal, $\vec{v}\cdot\vec{n}$. These normals are always taken as the outward facing normals. 
So, if you have a 1D control volume for example, the left side has a normal vector in the $-x$ direction and the right side has one in the $+x$ direction. If your velocity is in the $+x$ direction (meaning flow goes into the domain), then $\vec{v}\cdot\vec{n}$ on the left face is negative and on the right face is positive, so you get the signs you expect.
The other thing that would make it line up with your intuition is how you write it. The expression really is (ie. you didn't write an equation, you wrote a definition -- you need something on the RHS):
$$\frac{D B_{sys}}{Dt}=\frac{\partial B_{CV}}{\partial t}-\dot{B_{in}}+\dot{B_{out}} = S$$
where S is some source term (could be zero), which can be rearranged to be:
$$\frac{\partial B_{CV}}{\partial t} = \dot{B_{in}}-\dot{B_{out}} + S$$
and now it matches your intuition. Things flowing in will increase B in time, things flowing out will decrease it. 
