In fluid dynamics and continuum mechanics it is common to derive the the momentum balance by the following argument:
- For some open domain $V$, use the Reynolds Transport Theorem to show:
$$\frac{d\left(m\mathbf{v}_{CM}\right)}{dt}=\int_V \rho\frac{d\mathbf{v}}{dt}\,dV$$
- Assert Newton/Euler's law, where $A$ is the boundary of the region $V$:
$$\frac{d\left(m\mathbf{v}_{CM}\right)}{dt}=\int_A \mathbf{t}\,dA + \int_V \mathbf{f}\,dV $$
Prove Cauchy's Theorem: $$\mathbf{t}=\mathbf{T} \cdot \mathbf{n}$$
Substitute (3) into (2), use the divergence theorem, and equate the result with (1) to show that:
$$\int_V \rho\frac{d\mathbf{v}}{dt}\,dV=\int_V \left(\mathbf{f} + \nabla \cdot \mathbf{T}\right)\,dV $$
- Localize the integrand because the domain is arbitrary:
$$\rho\frac{d\mathbf{v}}{dt}=\mathbf{f} + \nabla \cdot \mathbf{T}$$
My question is why is it mathematically legitimate in step (2) to express the rate of change of momentum within the open domain $V$ in terms of a function defined on the boundary $A$, when by definition $A$ is outside the open domain? Is that type of expansion valid in general? Or is it only legitimate because we can also prove (3), so we're not "really" relating the rate of change within the domain to something outside of it, even though it appears that way in the standard derivation?
More generally, it seems like the momentum is a type of vector measure of the region $V$. Is the derivative of the measure of some open domain ever defined outside of that open domain?
