\[\oint_C dU + p dV\]

. \[U\]

is a function of \[p\]

and \[V\]

. We can define the state of a gas using these two coordinates, so a point \[(p,V)\]

determine \[U\]

uniquely.If

\[C\]

is a closed curve \[\oint_C dU =0\]

and the gas gains heat energy equal to \[\oint_C p dV\]

This is the area enclosed by the curve

\[C\]

.
If the curve \[C\]

is described clockwise then the area is taken as positive and the gas experiences a net gain of heat energy.