So, let me define a state function. A state function is a property that

depends only upon the state of the system.

It does not depend on how the system got to that state.

That is, it is independent of the path. And so, what defines a state, well we've

seen in some instances for instance, the specification of particular variables.

So we've worked with partition functions up 'til now for example that have

specified number of particles, temperature and volume.

And so a state function within that [UNKNOWN] would be something that depends

indeed only on number of particles, temperature and volume.

Not how I got there, not how I might have changed the temperature until I got to

the current temperature. A key property of a state function is

that its differential can be integrated as a mathematical quantity in a normal

path independent way. And so, in particular, energy is a state

function, internal energy. That is, I can think of integrating the

differential of the energy from state one to state two, that will be equal to the

difference between the energy of state two and state one, and I can write that

as just delta u. So, the energy difference, and it's

important to emphasize that in thermodynamics, we're almost always

interested in differences in quantities. It's quite rare we calculate something

absolutely, indeed, we often have to take conventions to define where zero is, and

that makes quite clear that we're usually interested in differences.

But, in any case we can integrate a differential for a state function to get

a change in a simple and path independent way.