# Conservation of Mass Question

Gold Member
Hi PF!

Can someone help me understand why, when writing the continuity equation we write: $$\frac{\partial}{\partial t} \iiint_v \rho \, dv$$ instead of $$\iiint_v \frac{\partial}{\partial t} \rho \, dv$$

I understand the two are not necessarily the same, but why derive it the first way rather than the second?

Intuitively, the first seems to be saying "add up all the mass and then see how it changes in time" where as the second seems to say "see how density changes in time at each location and then add it all up".

I'm just having trouble understanding the second integral.

Thanks!

Josh

Orodruin
Staff Emeritus
Homework Helper
Gold Member
If you keep your volume fixed, the two are equivalent. This is usually how you will see the continuity equation on differential form derived.

Gold Member
Yea, The same with energy and fluid balances. But I don't know what the second integral means, or rather why it's technically incorrect, from an intuitive perspective (mathematically I realize you need to use Leibniz' rule if the boundaries are time-dependent and you want to interchange the derivative and integral)

Any help on this is greatly appreciated.

Orodruin
Staff Emeritus