A differential of the form
![\begin{displaymath}
df=P(x,y)\,dx+Q(x,y)\,dy
\end{displaymath}](e_2564.gif) |
(1) |
is exact (also called a Total Differential) if
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}](e_2566.gif) |
(2) |
so
and
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}](e_2567.gif) |
(3) |
But
![\begin{displaymath}
{\partial P\over \partial y} = {\partial^2 f\over \partial y\partial x}
\end{displaymath}](e_2568.gif) |
(4) |
![\begin{displaymath}
{\partial Q\over \partial x}={\partial^2 f\over \partial x\partial y},
\end{displaymath}](e_2569.gif) |
(5) |
so
![\begin{displaymath}
{\partial P\over \partial y}={\partial Q\over \partial x}.
\end{displaymath}](e_2570.gif) |
(6) |
See also Pfaffian Form, Inexact Differential
© 1996-9 Eric W. Weisstein
1999-05-25