system
Given the following post, respond as if you're commenting on Reddit:
human
What does it actually mean if a quantity is an exact differential?

I came across this sentence:
 "since heat is not an exact differential it is not a property of the system. It is a path function."

So how can those things be inferred just by knowing that heat is not an exact differential?
gpt
The precise mathematical definition of an exact differential form is a differential form that can be written as an exterior derivative of another differential form.

In thermoynamics, the one-form dU (the differential internal energy) is an exterior derivative (d) of a zero-form (a function) U.

In other words, there is some function U that depends on the state of your thermodynamic system. Then taking the exterior derivative of that function gives you an exact one-form. A consequence of being an exact form is that the integral of the form is path-indepdent. So if you integrate dU over some path in the thermodynamic state-space, the result depends only on the initial and final states, and not on the path between them.

Now for an example of an inexact form: heat. Heat is fundamentally a **change** in energy during some process. It's denoted by ~~d~~Q, where the strikethrough denotes that it's an *inexact* one-form. There is no "heat function" Q (zero-form) whose exterior derivative is ~~d~~Q. You can't point at a thermodynamic system in some state and ask "How much heat does the system have?", that question is nonsense. So the differential heat cannot be written as an exterior derivative of a zero-form, so it's an inexact one-form. A consequence is that the integral of this one-form is path-dependent. During your thermodynamic process, if you move between the same initial and final points, but over different paths, the total amount of heat absorbed or released by your system will be different. It depends on the path you take.