Previous: 2.12 Maxwell's Relations
The rules of partial differentiation
Identify the independent variables, eg
and
.
If
, the partial derivative of
with respect to
is obtained by holding
constant;
it is written
It follows that
The order of differentiation doesn't matter:
The change in
as a result of changes in
and
is
![\begin{displaymath}
{\rm d}w=\left({\partial w\over\partial u}\right)_{\!\script...
...{\partial w\over\partial v}\right)_{\!\scriptstyle u} {\rm d}v
\end{displaymath}](img314.gif) |
(2.1) |
We could take
and
to be the independent variables, with
.
Now the partial derivatives are
Note the first is no longer required to be zero - it's
, not
, that is held constant. In this case,
![\begin{displaymath}
{\rm d}u=\left({\partial u\over\partial v}\right)_{\!\script...
...{\partial u\over\partial w}\right)_{\!\scriptstyle v} {\rm d}w
\end{displaymath}](img317.gif) |
(2.2) |
By comparing (2.1) and (2.2) with
, we see that
![\begin{displaymath}
\left({\partial w\over\partial u}\right)_{\!\scriptstyle v}=...
...aystyle{\partial u\over\partial w}}\right)_{\!\scriptstyle v}}
\end{displaymath}](img319.gif) |
(2.3) |
From (2.1) we have
![\begin{displaymath}
\mbox{\parbox{10cm}{
\begin{eqnarray*}
\left({\partial w\ove...
...\right)_{\!\scriptstyle u} \!\!\!&=&\!\!\!-1
\end{eqnarray*}}}
\end{displaymath}](img320.gif) |
(2.4) |
In the second line, ``dividing
by
'' gave
, not
, because the first line was only true for constant
.
Rearranging (2.4) also gives
![\begin{displaymath}
\left({\partial u\over\partial v}\right)_{\!\scriptstyle w} ...
...aystyle{\partial v\over\partial w}}\right)_{\!\scriptstyle u}}
\end{displaymath}](img325.gif) |
(2.5) |
The minus sign in these is counter-intuitive.
Equations (2.3), (2.4) and (2.5) are our main results, and may be new to you.
References
Previous: 2.12 Maxwell's Relations
Judith McGovern
2004-03-17