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I'm currently reading through a textbook on Real Analysis, and the following is written:

"The great advantage of Leibnitz symbols is that, when they are properly interpreted, we can treat the symbols d f, d y, dx, and so on, as if they represent members of IR and carry out various algebraic operations, and the resulting formulas will have meaning." - Patrick M Fitzgerald, Advanced Calculus, P114

In my physics courses, we very frequently say things like, "If you have a small amount of charge $dq$ in a small volume $dV$, we can define the charge density $\rho$ as $\frac{dq}{dV}$. However, I have been warned in the same classes that treatment of differentials in this manner is not mathematically rigorous and can lead to issues. What issues? I am not sure.

At the same time, in my Real Analysis course, we were told that $\frac{df(x)}{dx}$ should be treated as just notation, and not as a fraction.

And then I read the quoted text above, which appears to be merging the two concepts.

My question is this: How can differentials be treated? When is it rigorous to have a small amount of charge $dq$ be defined as $\rho dV$? My friends and I have grown skeptical of derivations of physics concepts when differentials were used, because we have been told to be skeptical. When is this warranted? When is it not? What do I watch for?

MurderOfCrows
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  • Part of the question might be a duplicate, but I'm interested in the part concerning the quote – vujazzman Nov 11 '19 at 20:51
  • Infinitesimals can be rigorously treated in the framework of hyperreal numbers (and generally non-standard analysis), but I don't know if that's what Fitzgerald was aiming at. – Randy Marsh Nov 11 '19 at 22:20

1 Answers1

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From an overview, derivatives are just describing a change. For example, the derivative $ \frac{dy}{dx} $ for some equation (say y = 5x + 6 is 5) is 5 since y changes +5 units for each increasing x value. When you see fractions with derivatives on the top and bottom, this equates to "change in y over the change in x" meaning that you are taking the ratio of the number of values y changes for each value of x. For the equation above, you move 5 units up for every unit you move to the right (aka "rise" over "run"). Since y = f(x), sometimes $ \frac{dy}{dx} $ is written as $ \frac{df(x)}{dx} $. When derivatives are first introduced, $ \frac{dy}{dx} $ is rarely separated so often times it is considered a single entity.

Using your example, $\rho = \frac{d\rho}{dV}$, we can say that $\rho$ is equal to the ratio of the change in $\rho$ over the change in $V$. Basically equate $d$ to "change." To make things more confusing, physicists often use $ y' $ instead of $ \frac{dy}{dx} $. Leibnitz used $\frac{dy}{dx}$ instead since in higher level math courses you need to separate the fraction $\frac{dy}{dx}$. Hopefully this helps to explain derivatives a bit.

suhbell
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