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### Things not to do with Differentials

Differentials are a wonderful tool for manipulating derivatives. However, it is important to remember that differentials themselves always refer to the *total* change in a quantity. Ratios of differentials can often be interpreted as ordinary derivatives, but not as partial derivatives. Put differently, correct statements about differentials can be obtained by pulling apart an *ordinary* derivative, but never by pulling apart a *partial* derivative.

For instance, it is fine to convert the chain rule statement \begin{equation} \frac{df}{dt} = \Partial{f}{x}\frac{dx}{dt} + \Partial{f}{y}\frac{dy}{dt} \label{chainder} \end{equation} to the statement \begin{equation} df = \Partial{f}{x} \,dx + \Partial{f}{y} \,dy \label{chaindiff} \end{equation} by “multiplying” both sides by $dt$. In fact, we like to start with ($\ref{chaindiff}$) and obtain the usual chain rule statement ($\ref{chainder}$) by “dividing” both sides by $dt$. However, this can not be done with partial derivatives.

Consider for example the partial differential equation \begin{equation} \Partial{u}{x} = x \Partial{u}{y} \label{diffex} \end{equation} One way to obtain a solution of ($\ref{diffex}$) is by separation of variables: 1) \begin{eqnarray} u(x,y) &=& X(x) Y(y) \\ \Longrightarrow && \frac{1}{xX}\,\frac{dX}{dx} = \hbox{constant} = \frac{1}{Y}\,\frac{dY}{dy} \\ \Longrightarrow && \frac{dX}{X} = cx\,dx \qquad\hbox{and}\qquad \frac{dY}{Y} = c\,dy \\ \Longrightarrow && u = A\, e^{c(y+\frac12 x^2)} \end{eqnarray}

Contrast this correct use of differentials with the following, incorrect, argument. Rewrite ($\ref{diffex}$) as \begin{equation} du\,dy = x\,du\,dx \end{equation} Now cancel $du$ from both sides, obtaining \begin{equation} dy = x\,dx \end{equation} suggesting that the “solution” to ($\ref{diffex}$) is given by \begin{equation} y = \frac12 x^2 \end{equation}

The moral is that *partial* derivatives can not be treated as ratios of differentials. Do not be misled by ($\ref{chaindiff}$) itself, which does indeed imply that \begin{equation} df = \Partial{f}{x} \,dx \label{diffshort} \end{equation} *if* $y=\hbox{constant}$; that assumption effectively turns the partial derivative into an ordinary derivative. If several variables are changing, “shortcuts” such as ($\ref{diffshort}$) are not valid.