Notion of independence in thermodynamical systems (introductory) I'm trying to self-study thermodynamics and these notes stated the state postulate (two independent properties are sufficient to describe the full state) and then hand-wavily noted that pressure and density are always independent. 
Does this simply mean that knowing these two are always sufficient to describe the whole system? If so, it seems like a very poetic way to use the notion of "independence". Upon reading that statement, I would have inferred that $\frac{\partial P}{\partial \rho} = 0 = \frac{\partial \rho}{\partial P}$. But this sounds intuitively absurd! Of course the pressure must change if i increase the density of the substance, right?
 A: The temperature, pressure, and density are inter-related.  If you change the density, the pressure can be held constant by adjusting the temperature.  Usually, the two independent properties chosen are pressure and temperature, rather than pressure and density.
A: As an example, let's take a look at a well-known equation-of-state, the ideal-gas law:
$$ pv = rT $$
with pressure $p$, specific volume $v$, temperature $T$ and specific gas constant $r$, or $$\frac{p}{\rho} = rT $$ if you prefer working with density, $\rho = 1/v$.
This equation can be put in the following form:
$$ F(p,v,T) = pv-rT = 0 $$
It should now be obvious that if you know 2 of the 3 state variables, the third can be calculated from solving $F(p,v,T) = 0$. Similar equations-of-state exist for every material (often not in analytical form). One point of attention is that some state variables are not always dependent. E.g. a pot of boiling water at given pressure and temperature is underdetermined, because it doesn't specify how much liquid and how much vapor there is. This is because the boiling temperature at a given pressure is fixed, and thus $p$ and $T$ are dependent. So, it is up to you to pick the independent variables, but you should be carefull.
Now, what's the problem with your intuition? Starting from the assumption that $p$ and $\rho$ are independent variables describing the state of the system, it is meaningless to ask what is $\frac{\partial p}{\partial \rho}$. Actually, it is meaningless regardless of the choice of independent parameters. It is important in thermodynamics to specify what the process is, when calculating partial derivatives. You may be imagining an isothermal process, but somebody could be thinking about an adiabatic process. Then both of you would end up with a different $\frac{\partial p}{\partial \rho}$. And you would both be "right" to some extend, i.e. both your answers would be incomplete without specifying the process.
A: By "independence" is meant that in an experiment, you can independently vary pressure and density.
