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.

1) In the absence of boundary conditions, this approach would yield a general solution of the form \begin{eqnarray*} u = \int A( c )\, e^{c(y+\frac12 x^2)} \,dc \end{eqnarray*} in agreement with the “obvious” solution $u=f(y+\frac12x^2)$ (which is however not in general separable).