First rocket equation (sign before $dm$) I'm trying to understand the first rocket equation. At start it says:
$$P(t+dt)-P(t)=dP$$
$$((m+dm)(v+dv)-dm(v+dv+v_r))-mv=Fdt$$
I understand why there is minus before $dm$ at the second part but what I don't understand is why there is no minus before $dm$ in $m+dm$.
How can mass of the rocket increase when the rocket is consuming fuel?
 A: dm is defined to be the infintesimal change in the rocket's mass. In math:
$$
dm = m(t+dt)-m(t)
$$
If the rocket is consuming fuel, dm is negative. If the rocket was somehow spontaneously creating fuel, dm would be positive. In either case, the mass after time $dt$ is $m+dm$. It's never $m-dm$. The minus sign is absorbed in the $dm$.
In general, dx is just the change in x. dx can be positive or negative. You don't explicitly put in the sign, because it's implicit in the definition of dx.
If you like, you could simplify the expression you wrote above and solve for $dm$. Once you do this, you'll see explicitly that $dm$ is negative, just like you'd expect.
A: I think another approach to the rocket equation is better. We assume that we have a rocket that is moving with momentum $p~=~mv$. This rocket ejects a small increment of mass $\Delta m$ at velocity $-V$ relative to the rocket or $-V~+~v$ to the lab frame. The rocket mass is reduced to $m~-~\Delta m$ and its velocity increases to $v~+~\Delta v$ as a result the total momentum conserved is
$$
mv~=~(m~-~\Delta m)(v~+~\Delta v)~-~(V~-~v)\Delta m.
$$
Now do the mathematics and ignore the $\Delta m\Delta v$ term as negligible so
$$
m\Delta v~=~-V\Delta m.
$$
Now divide the m and integrated
$$
\int_{v_i}^{v_f}dv~=~v_f~-~v_i~=~-V\int_{m_i}^{m_f}\frac{dm}{m} = V~ln\left(\frac{m_i}{m_f}\right).
$$
This is why if a rocket puts $10\%$ of its mass in orbit that it reaches terminal velocity $2.3V$, where $V$ is the velocity of the gas escaping the rocket. A top chemical rocket has a specific impulse $isp~=~v/g$ $=~550s$ for $g~=~9.8m/s^2$ This means the plume velocity is $5400m/s$ and so the terminal velocity is $12400m/s$. 
