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Homework Statement
Show that
[itex]\frac{d}{dt}\int \rho r^{2}\phi dr = \int \rho r^{2}\frac{d\phi}{dr} dr [/itex]
Homework Equations
Fundamental theorem of calculus
The Attempt at a Solution
So I follow the derivation from the textbook and I think I get the rather sneaky rearrangement of the derivatives, but I do not see how
[itex]\int \rho r^{2}\frac{d\phi}{dt} dr = \int \rho r^{2}\left(\frac{\partial \phi}{\partial t}+v\frac{\partial \phi}{\partial r}\right)dr [/itex]
Note: Integrals are evaluated from a to b, and v(x,t) = dx/dt (e.g. da/dt = v(a,t))