Section 7.5 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
to the statement
by “multiplying” both sides by \(dt\text{.}\) In fact, we like to start with (7.5.2) and obtain the usual chain rule statement (7.5.1) by “dividing” both sides by \(dt\text{.}\) However, this can not be done with partial derivatives.
Consider for example the partial differential equation
One way to obtain a solution of (7.5.3) is by separation of variables. In the absence of boundary conditions, this approach would yield a general solution of the form
in agreement with the “obvious” solution \(u=f(y+\frac12x^2)\) (which is however not in general separable):
Contrast this correct use of differentials with the following, incorrect, argument. Rewrite (7.5.3) as
Now cancel \(du\) from both sides, obtaining
suggesting that the “solution” to (7.5.3) is given by
The moral is that partial derivatives can not be treated as ratios of differentials. Do not be misled by (7.5.2) itself, which does indeed imply that
if \(y=\hbox{constant}\text{;}\) that assumption effectively turns the partial derivative into an ordinary derivative. If several variables are changing, “shortcuts” such as (7.5.7) are not valid.