The rules of partial differentiation

$\bullet$ Identify the independent variables, eg $u$ and $v$.
$\bullet$ If $w=w(u,v)$, the partial derivative of $w$ with respect to $u$ is obtained by holding $v$ constant; it is written

$$\left({\partial w\over\partial u}\right)_{\!\scriptstyle v}$$

$\bullet$ It follows that

$$\left({\partial v\over\partial u}\right)_{\!\scriptstyle v} ... ...=\left({\partial v\over\partial v}\right)_{\!\scriptstyle u}=1$$

$\bullet$ The order of differentiation doesn't matter:

$$\left({\partial\over\partial v}\left({\partial w\over\partia... ...partial v}\right)_{\!\scriptstyle u}\right)_{\!\scriptstyle v}$$

$\bullet$ The change in $w$ as a result of changes in $u$ and $v$ is

$${\rm d}w=\left({\partial w\over\partial u}\right)_{\!\script... ...{\partial w\over\partial v}\right)_{\!\scriptstyle u} {\rm d}v$$ (2.1)

$\bullet$ We could take $v$ and $w$ to be the independent variables, with $u=u(v,w)$. Now the partial derivatives are

$$\left({\partial u\over\partial v}\right)_{\!\scriptstyle w}\... ...uad\left({\partial u\over\partial w}\right)_{\!\scriptstyle v}$$

Note the first is no longer required to be zero - it's $w$, not $u$, that is held constant. In this case,

$${\rm d}u=\left({\partial u\over\partial v}\right)_{\!\script... ...{\partial u\over\partial w}\right)_{\!\scriptstyle v} {\rm d}w$$ (2.2)

$\bullet$ By comparing (2.1) and (2.2) with ${\rm d}v=0$, we see that

$$\left({\partial w\over\partial u}\right)_{\!\scriptstyle v}=... ...aystyle{\partial u\over\partial w}}\right)_{\!\scriptstyle v}}$$ (2.3)

$\bullet$ From (2.1) we have

$$\mbox{\parbox{10cm}{ \begin{eqnarray*} \left({\partial w\ove... ...\right)_{\!\scriptstyle u} \!\!\!&=&\!\!\!-1 \end{eqnarray*}}}$$ (2.4)

In the second line, ``dividing ${\rm d}u$ by ${\rm d}v$'' gave $\left({\partial u\over\partial v}\right)_{w}$, not ${{\rm d}u\over {\rm d}v}$, because the first line was only true for constant $w$.
$\bullet$ Rearranging (2.4) also gives

$$\left({\partial u\over\partial v}\right)_{\!\scriptstyle w} ... ...aystyle{\partial v\over\partial w}}\right)_{\!\scriptstyle u}}$$ (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

• Bowley and Sánchez E.1