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### Rearranging Differentials

Using differentials allows algebraic operations to yield information about differentiation. Not only do we know that \begin{equation} df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy \end{equation} but we can run this argument in reverse.

Suppose we know that \begin{equation} du = A\,dv + B\,dw \end{equation} Then we also know that \begin{eqnarray} A &=& \Partial{u}{v} \\ B &=& \Partial{u}{w} \end{eqnarray} Furthermore, we can use algebra to solve for $dv$, obtaining \begin{equation} dv = \frac{1}{A}\,du - \frac{B}{A}\,dw \end{equation} and we can conclude that \begin{eqnarray} \Partial{v}{u} &=& \frac{1}{A} \\ \Partial{v}{w} &=& -\frac{B}{A} \end{eqnarray}

With so many variables in use at the same time, it is important to specify which ones are being held constant when taking derivatives; this is usually done by writing $\left(\Partial{v}{u}\right)_w$ to denote the partial derivative of $v$ with respect to $u$ *with $w$ held constant*. We have therefore shown that \begin{eqnarray} \left(\Partial{u}{v}\right)_w \left(\Partial{v}{u}\right)_w &=& 1 \\ \left(\Partial{u}{v}\right)_w \left(\Partial{v}{w}\right)_u + \left(\Partial{u}{w}\right)_v &=& 0 \end{eqnarray} and the latter of these equations is usually rewritten in the form \begin{equation} \left(\Partial{u}{v}\right)_w \left(\Partial{v}{w}\right)_u \left(\Partial{w}{u}\right)_v = -1 \end{equation} and called the *cyclic chain rule*.

In practice, using algebra to rearrange equations involving differentials automatically incorporates the chain rule in all of these forms. It is often easier to rearrange than to use the formulas.