Energy is a function of the state of the system, not how you get there. This is why it is important to be clear on the nomenclature, and to recognize various terms as indicators of the assumptions that have been made. In recitation tomorrow, I will give you a bit of a First Law equation roadmap to help with this. Don't understand why work is path independent if the process is adiabatic. This idea is expressed in one of the corollaries of the First Law .
Internal energy is a function of the thermodynamic state of the system.
It does not depend on the process by which the system arrived at that state. W. So work is equal to something that does not depend on path. The differential forms of the First Law were not well explained. We will talk more about these in recitation tomorrow. The differential forms of the First Law are just a useful, convenient way to write them that sometimes make manipulating the equations easier.