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.

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