Exact Differential

A differential of the form

$$df=P(x,y)\,dx+Q(x,y)\,dy$$ (1)

is exact (also called a Total Differential) if $\int df$ is path-independent. This will be true if

$$df={\partial f\over \partial x}\,dx+{\partial f\over \partial y}\,dy,$$ (2)

so $P$ and $Q$ must be of the form

$$P(x,y)={\partial f\over \partial x} \qquad Q(x,y)={\partial f\over \partial y}.$$ (3)

But

$${\partial P\over \partial y} = {\partial^2 f\over \partial y\partial x}$$ (4)

$${\partial Q\over \partial x}={\partial^2 f\over \partial x\partial y},$$ (5)

so

$${\partial P\over \partial y}={\partial Q\over \partial x}.$$ (6)

See also Pfaffian Form, Inexact Differential