What is the true, formal meaning and reason for the "dx" symbol in integrals

Question

When I encountered integrals for the first time, and learned to write the "dx" at the end of every integral, I had no problem interpreting it as something that told me what the variable of integration is, or where the integral ends, and nothing more. But when I encountered u-substitution, we started doing things like du = u'(x)dx, and replacing u'(x) and dx with du in the integral. Well, that seems like "dx" was never just a delimiter, but something being multiplied with the function itself. I asked around about this, and people told me that dx is in fact, only a delimiter to tell us our variable of integration, and that the "multiplication" I did was just some sort of mnemonic for the reverse chain rule. I thought it was weird to use a mnemonic like that, but I understood it. But then others told me that "dx" is part of what's being integrated, and they started saying that we're led to believe that its just a delimiter in early courses because it'd be impossible for teachers to introduce "differentials," which is what things like dx and du are, so u-substitution isn't just a mnemonic, and the multiplication is completely formal. They also said that I haven't been integrating functions, but rather differential forms, and have only been told I'm integrating functions to make things easier until I learn the truth.

This is all extremely confusing to me. I have no idea how I've heard so many differing opinions that can't be true at the same time. This all, once again, leaves me wondering, what is the real, formal meaning of the notation we use for integrals, what does that "dx" truly represent - is it a part of the computation, or is it something easily replaceable by a string like "with respect to x"? Do we integrate functions, or do we integrate something called a differential form? How much of what I've been told is true, and what haven't I been told? This has been bothering me for some time, so I'd greatly appreciate it if anyone could try to clear this up for me!

Answer 1

It depends on whom you ask. Prior to the 1800s, "dx" was considered an "infinitesimal" - a number so close to zero that, for some things, it can be considered actually zero, but wasn't exactly zero.

In the 1800s, the failure to formalize infinitesimals (and, in my opinion, the growing rise of materialism) led to the belief that infinitesimals were invalid mathematical objects. This led to the use of limits as the foundation of calculus.

However, in the 1960s, the infinitesimals were finally formalized, giving "dx" a real foundation as an infinitesimal number.

The way to think about $dy$ and $dx$ is that they are infinitely small values. $dy$ and $dx$ are both infinitely small, but they are distinct. The derivative (or other differential equation) tells you the relationship between $x$, $y$, $dx$, and $dy$.

Let's say you have the equation $y = x^2$. The derivative is $\frac{dy}{dx} = 2x$. Or, in terms of differentials, you could write that as $dy = 2x\,dx$. The way to interpret that is to say, whatever infinitely small increment that $x$ is changing (i.e., $dx$), the amount that the infinitesimal $dy$ is changing is that same infinitesimal multiplied by $2x$, wherever you happen to be on the $x$ axis.

Anyway, while equations involving $x$ and $y$ alone tell you about the relationship between the values of the variables, equations involving $dx$ and $dy$ tell you about the relationship between the changes that are occurring in the variables.

Answer 2

$x:M\to \mathbb R$ is a coordinate function; $dx_p:TM_p\to T\mathbb R_p=\mathbb R$ is the corresponding map derivative at a point $p\in M$ -- it acts on tangent vectors; Integral operator feeds $dx$ (infinitesimal) tangent vectors at (continuum) consecutive points along a curve: $\int_C dx=\lim\sum dx_{p_i}(v_i)$.