What is state variable and full differential? e.g. in entropy? I am studying basic concepts of entropy and statistical physics. And red a lot what is entropy, and that it is integrative factor of heat; and getting it with the full differential._
Anyway, what I am trying to grasp in all that story, actually to gain a feeling and know what is actually full differential means in this story; and that some variable is state variable (state function)?
For example $dS=\delta Q/dT$.
I know I needed to find integrative factor which is temperature as universal thermodynamics variable. And I need to have on extensive and one intensive variable to get state variable.
But get confused if I actually just start to think about what is variable of state here actually and what does it usually say.
It same here with work where $dV= \delta W/ -p$.
Thanks for help :)
 A: A function of state is one that depends only on the state of the system, which for a gas means that it's a function of the pressure $p$, volume $V$, and temperature $T$, and not dependent on the path by which the system got there. They're also only defined in equilibrium; if there's no defined $p$, $V$ or $T$, there's no defined functions of them.
For example, internal energy is a function of state - no matter how your gas got to be in that box at that temperature and pressure, it's always going to contain the same amount of energy. No matter how it's compressed, expanded, cooled down, or heated up, if it goes back to the same pressure, volume and temperature, it goes back to the same internal energy.
Work, on the other hand, isn't (for example). The amount of work done on or by a gas depends on the specific path that it takes through pressure, volume and temperature-space. Different thermodynamic cycles will, in general, have different amounts of work associated with them.
A: There are a lot of resources out there to help you. A good complete introductory resource is Mark Zemansky’s text book “Heat and Thermodynamics”. I believe you can access the 7th edition on line.
A differential change in entropy is defined for a reversible, differential transfer of heat divided by the temperature at which it is transferred per the following equation:
$$ds= \frac {dq_{rev} }{T}$$ 
But you need to be careful not to think that heat transfer is required for entropy change. You can have a change in entropy without any heat transfer. An example is an adiabatic (Q=0) free expansion of a gas. 
Entropy is a property, or state function. The same applies to Pressure, Temperature, Volume, Internal Energy, Enthalpy, etc. This means, that a change in entropy of a system between two equilibrium states is independent of the process or path between the states. So while it may be defined for a reversible heat transfer process, any path or process(es) that takes you between the two states will have the same system entropy change (though the change in the entropy of the surroundings will be different unless the process is reversible).
Work (as well as heat), on the other hand is not a property of a system. The amount of work done or heat transferred in going from one equilibrium state to another depends on the process (path) between the states.
This is just a start. 
Hope this helps and good luck!
A: The equation should read $dS=dQ_{rev}/T$, where the subscript rev refers exclusively to a reversible path between the initial and final states of the system. This reversible path does not necessarily have to bear any resemblance whatsoever to the actual process path between the initial and final states. If you don't apply the equation to a reversible path (which you may have to devise), the equation does not give the entropy change. Only for a reversible path is 1/T an integrating factor for dQ to obtain dS.  For a primer on how to determine the entropy change for a system experiencing any process (whether reversible or irreversible), see the following link:  https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe/
