Theory of exact differentials

Theory of exact differentials

A differential, $\textrm{d}f$, is said to be exact if: $$ \oint \textrm{d}f = 0 $$ An exact differential, $\textrm{d}f=C_1(x,y)\textrm{d}x + C_2(x,y)\textrm{d}x$, has the following important property: $$ \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} $$ Therefore: $$ \left(\frac{\partial C_1}{\partial y}\right)_x = \left(\frac{\partial C_2}{\partial x}\right)_y $$ which follows by remembering that $\textrm{d}f = \left(\frac{\partial f}{\partial x}\right)_y \textrm{d}x + \left(\frac{\partial f}{\partial y}\right)_x \textrm{d}y$ for an exact differential. Two final useful results can be obtained by comparing coefficients of $\textrm{d}x$ and $\textrm{d}y$ in $\textrm{d}f=C_1(x,y)\textrm{d}x + C_2(x,y)\textrm{d}y$ and $\textrm{d}f = \left(\frac{\partial f}{\partial x}\right)_y \textrm{d}x + \left(\frac{\partial f}{\partial y}\right)_x \textrm{d}y$.

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