- #1
member 428835
hey pf!
i had a question. namely, in the continuity equation we see that [tex] \frac{\partial}{\partial t}\iiint_V \rho dV = -\iint_{S} \rho \vec{v} \cdot d\vec{S}[/tex] and we may use the divergence theorem to have: [tex] \frac{\partial}{\partial t}\iiint_V \rho dV = -\iiint_{V} \nabla \cdot \big( \rho \vec{v} \big) dV[/tex]
ultimately, we arrive at: [tex] \frac{\partial}{\partial t}\bigg( \rho \bigg) = -\nabla \cdot \big( \rho \vec{v} \big)[/tex]
my question is, at this point, how are we able to do the following two things:
1 interchange ##\frac{\partial}{\partial t}## inside the volume integral?
2 drop the volume integrals entirely?
i should say that the volume is arbitrary, and from what i remember, we have to do something like ##\lim_{V \to 0}## but i don't know the formal, mathematical procedure here.
can someone please help me out?
thanks!
i had a question. namely, in the continuity equation we see that [tex] \frac{\partial}{\partial t}\iiint_V \rho dV = -\iint_{S} \rho \vec{v} \cdot d\vec{S}[/tex] and we may use the divergence theorem to have: [tex] \frac{\partial}{\partial t}\iiint_V \rho dV = -\iiint_{V} \nabla \cdot \big( \rho \vec{v} \big) dV[/tex]
ultimately, we arrive at: [tex] \frac{\partial}{\partial t}\bigg( \rho \bigg) = -\nabla \cdot \big( \rho \vec{v} \big)[/tex]
my question is, at this point, how are we able to do the following two things:
1 interchange ##\frac{\partial}{\partial t}## inside the volume integral?
2 drop the volume integrals entirely?
i should say that the volume is arbitrary, and from what i remember, we have to do something like ##\lim_{V \to 0}## but i don't know the formal, mathematical procedure here.
can someone please help me out?
thanks!