In my Calculus class, my math teacher said that differentials such as $dx$ are not numbers, and should not be treated as such.
In my physics class, it seems like we treat differentials exactly like numbers, and my physics teacher even said that they are in essence very small numbers.
Can someone give me an explanation which satisfies both classes, or do I just have to accept that the differentials are treated differently in different courses?
P.S. I took Calculus 2 so please try to keep the answers around that level.
P.S.S. Feel free to edit the tags if you think it is appropriate.
Answer
(I'm addressing this from the point of view of standard analysis)
I don't think you will have a satisfactory understanding of this until you go to multivariable calculus, because in calculus 2 it's easy to think that $\frac{d}{dx}$ is all you need and that there's no need for $\frac{\partial}{\partial x}$ (This is false and it has to do with why in general derivatives do not always behave like fractions). So that's one reason why differentials are not like numbers. There are some ways that differentials are like numbers, however.
I think the most fundamental bit is that if you're told that $f dx=dy$, this means that $y$ can be approximated as $y(x)=y(x_0)+f\cdot(x-x_0)+O((x-x_0)^2)$ close to the point $x_0$ (this raises another issue*). Since this first order term is really all that matters after one applies the limiting procedures of calculus, this gives an argument for why such inappropriate treatment of differentials is allowable - higher order terms don't matter. This is a consequence of Taylor's theorem, and it is what allows your physics teacher to treat differentials as very small numbers, because $x-x_0$ is like your "dx" and it IS a real number. What allows you to do things you can't do with a single real number is that that formula for $y(x)$ holds for all $x$, not just some x. This lets you apply all the complicated tricks of analysis.
If I get particularly annoyed at improper treatment of differentials and I see someone working through an example where they write, "Now we take the differential of $x^2+x$ giving us $(2x+1)dx$", I may imagine $dx$ being a standard real number, and that there's a little $+O(dx^2)$ tacked off to the side.
Your math teacher might argue, "You don't know enough about those theorems to apply them properly, so that's why you can't think of differentials as similar to numbers", while your physics teacher might argue, "The intuition is the really important bit, and you'd have to learn complicated math to see it as $O(dx^2)$. Better to focus on the intuition."
I hope I cleared things up instead of making them seem more complicated.
*(The O notation is another can of worms and can also be used improperly. Using the linked notation I am saying "$y(x)-y(x_0)-f\cdot(x-x_0)=O((x-x_0)^2)$ as $x\to x_0$". Note that one could see this as working against my argument - It's meaningless to say "one value of $x$ satisfies this equation", so when written in this form (which your physics prof. might find more obtuse and your math prof. might find more meaningful) it's less of an equation and more of a logical statement.)
See also: https://mathoverflow.net/questions/25054/different-ways-of-thinking-about-the-derivative
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