Take-home message: Remember this equation!
The first law for infinitesimal changes says . Since it is obviously true for reversible changes, we have . We have already found expressions for reversible work for a variety of systems; now we have one for reversible heat transfer too: . So we can put these together to form an expression for which only involves functions of state. For a hydrodynamic system, for instance,
For other systems, is replaced by the appropriate expression for , eg for a stretched string.
The significance of this equation should become clear as the course continues: it is one of the most important half-dozen equations we will meet.