Mass conservation

Question

I am trying to prove that $$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \int_{a(t)}^{b(t)} \rho(x, t)\frac{D}{Dt}g(x, t)dx$$

I have tried to evaluate the integral using Liebniz' rule, so that

$$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \int_{a(t)}^{b(t)}\left[\rho(x, t)g(x, t)\right]_t \,dx$$

but am unsure if the rule can be applied in this manner. Would I then apply some manner of multivariate product rule?

Any help much appreciated,

Will

Answer 1

You can apply the chain rule to the fundamental theorem of calculus. You get $$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \frac {db(t)}{dt}\rho(x, t)g(x, t)-\frac {da(t)}{dt}\rho(x, t)g(x, t)$$ Maybe that helps.

Answer 2

This is reynolds transport theorem. You are missing alot of assumptions here.