# Deriving a conservation law using the divergence theorem

Problem:

Material scientists have discovered a new fluid property called "radost" that is carried along with a fluid as it moves from one place to the next (just like a fluid's mass or momentum). Let ##r(x,y,z,t)## be the amount of radost/unit mass in a fluid. Let ##\rho(x,y,z,t)## be the mass density of the fluid. Let ##\vec{v}(x,y,z,t)## be the velocity vector of the fluid. Use the divergence theorem to derive a conservation law for radost.

Attempt at a Solution:

We did an example like this in class, but for conserving mass, so it was a little different. What we ended up with was the following expression $$\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v})=0$$

We started by writing, $$dM=\rho dV$$ Thus, $$M=\int_V \rho dV$$ Then we applied the divergence theorem and that was basically it.

I'm just kind of confused how to start this one.

Office_Shredder
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