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I've done a bit of searching and found a fairly well written explanation, but at the end, the author noted that this explanation seems to work fine for a physicist's purposes - but a mathematician would groan at it due to oversimplifications or inaccuracies.

Since I first posted this paper, two different people have emailed me to tell me that Real Mathematicians don't do this. Playing with dx in the ways described in this paper is apparently one of those smarmy tricks that physicists use to give headaches to mathematicians.

I was also confused when reading it because my Calculus prof last semester said that the chain rule

$$\frac{dy}{dz}=\frac{dy}{dx}\cdot \frac{dx}{dz}$$

cannot be treated as fractions, despite the fact that they look like it, and the $dx$ would not cancel out between the two since you can't do that with differentials. But this article said just the opposite.

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You can think of it as just a symbol to say that your integrating function is a function of the variable $x$. So, you know what I'm saying if I write $\int x^2y+\cos(yx) dx$. – Sigur Jan 19 '13 at 16:18
duplicate of – Zev Chonoles Jan 19 '13 at 16:23
Your calculus professor is right. A lot of these little "tricks", however, are allowed in a branch of mathematics known as non-standard analysis. – George V. Williams Jan 19 '13 at 16:25
@ZevChonoles: I disagree with closing this as a duplicate since this question seems to contrast the physicists and the mathematicians way of understanding $dx$. I think something useful can be said here. – Thomas Jan 19 '13 at 16:47
up vote 10 down vote accepted

I have heard from several mathematicians that one thing that they didn't like about physics was the "abuse" of mathematics. Physicists are not necessarily bad mathematicians, they (IMO) just use math differently. For them, the precise nature isn't (always) at the core of what they are doing. Since the physicist works with the "real world", he/she doesn't often work with approximations and therefore is not passionately interested in the exact nature of equations and such (hopefully I didn't offend anyone).

I see on the site that you give reference to, that the person thinks about $dx$ as a finite quantity. He talks for example about $x + dx$ as being a real number. This is not necessarily a bad way to think about it, but again it isn't the exact definition.

About the integral: When we write for example $$ \int_a^b f(x)\; dx $$ we have defined this symbol as is. We haven't defined the symbol as made up of different individual component that can we written on their own. So even though $f(x)$ in the expression indeed does have meaning on its own, we are not allowed to remove it and just write: $$ \int_a^b dx. $$ Now, someone might actually use this notation from time to time, but in "standard basic calculus" this hasn't been defined, and so it doesn't have any meaning. (It doesn't mean that we can think about it in a certain way).

Now all that said, this doesn't mean that there isn't a rationale behind the choice of the different components of the symbol. So, of course, the $f(x)$ is the function. The $a$ and the $b$ are the limits of the integral and the $dx$ tells us to treat $x$ as the variable (we say that we integrate with respect to $x$). Why is this important? As mentioned in another answer, if you consider the integral: $$ \int_{a}^{b} x^2y. $$ you wouldn't know if $x$ is the variable of $y$ is the variable (or if both are constants). So that is why we like to "add" the $dx$ or $dy$ to the end.

This isn't the only reason. The definite integral $$ \int_a^b f(x)\; dx $$ is defined as a limit: $$ \lim_{n\to \infty}\sum_{i=1}^{n} f(x_i)\Delta x. $$ Here $\Delta x$ is a finite number defined as $\Delta x = \frac{b - a}{n}$. And taking the limit $n\to \infty$ we have $\Delta x \to 0$. So we "think" of $dx$ as this infinitely small number. We can "think" of the $\Delta$ as becomming the $d$, even though the $dx$ can't be written on its own.

The same applies to the derivative $$ \frac{dx}{dy}. $$ Here again we have just defined this one symbol where the parts of the symbol ($dx$ and $dy$) are not well defined on their own.

All this said, one can actually make a meaning (i.e. define) of $dx$, but when we are just taking about basic calculus, we don't usually do that.

Also note that some mathematicians will just write $\int f(x)$ when they mean $\int f(x) \; dx$ because it is "obvious" that we are integrating with respect to $x$.

You can find more things about the $dx$ in this question: What is $dx$ in integration?

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A possibility is to see $dx$ as a differential form. Then, the integration for differential forms is well defined.

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There are a number of ways to view the $dx$. As Seirios pointed out it might be a differential form, which is the usual way to have it "make sense.", but it's often simply defined as the notation of an integral. Integrating a function $f$ with some measure $\mu$ can be described by the notation $\int f d\mu$ and when the measure used is the Lebesgue measure it is customary to replace $d\mu$ with $dx$. In this way the notation "makes sense" because it is well defined, but the $dx$ itself has no meaning without the explicit context of the integral.

$\frac{dy}{dx}$ means something entirely different, that is the derivative of $y$ with respect to $x$. The reuse of the symbols, and the fact that $\frac{dy}{dx}$ looks like a fraction is only because the symbols were in use long before the appropriate formalism was invented for calculus to make sense rigorously, and at that point there was no turning back.

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Remember, that from a historical perspective, differential calculus was developed first from the "physics" point of view, that is, $\int f(x) dx$ was originally thought of as a sum (the $\int$ sign is actually an $s$): $$\sum f(x_i)\Delta x$$ Similarly, the chain rule makes a lot of sense if used as a fraction, and that was it's original meaning. The problem is that when trying to set mathematics on a rigorous foundation, one could no longer use the original "intuitive" definitions. These "physical" methods help us think about what the definitions mean, but we cannot use them in mathematical proofs.

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