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Exact Differential

A differential of the form

\begin{displaymath}
df=P(x,y)\,dx+Q(x,y)\,dy
\end{displaymath} (1)

is exact (also called a Total Differential) if $\int df$ is path-independent. This will be true if
\begin{displaymath}
df={\partial f\over \partial x}\,dx+{\partial f\over \partial y}\,dy,
\end{displaymath} (2)

so $P$ and $Q$ must be of the form
\begin{displaymath}
P(x,y)={\partial f\over \partial x} \qquad Q(x,y)={\partial f\over \partial y}.
\end{displaymath} (3)

But
\begin{displaymath}
{\partial P\over \partial y} = {\partial^2 f\over \partial y\partial x}
\end{displaymath} (4)


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

so
\begin{displaymath}
{\partial P\over \partial y}={\partial Q\over \partial x}.
\end{displaymath} (6)

See also Pfaffian Form, Inexact Differential




© 1996-9 Eric W. Weisstein
1999-05-25