I am a physics student learning more about math (and thinking of switching my major to math). I am trying to learn more about how calculus is looked at through the rigorous lens of a mathematician, and it is my understanding that mathematicians no longer think of notation such as $dx$ as representing an "infinitesimal quantity," but instead, they think of calculus in terms of limits. However, we all know that when integrating, we must not forget the $dx$ (or whatever variable we are integrating with respect to). I have always been told that this $dx$ is here because we are multiplying the function at each point by this infinitesimal change in $x$. However, this only seems to make sense to me in the context of viewing $dx$ as an infinitesimal, and I am not sure how this can make sense if we view calculus as being "about" limits instead of infinitesimals. Is this outdated notation, or am I missing/lacking knowledge of something? Thank you.
-
5$dx$ can be understood as a differential form or a measure, see https://en.wikipedia.org/wiki/Differential_form, https://en.wikipedia.org/wiki/Lebesgue_integral. – Qiaochu Yuan Mar 27 '25 at 22:04
-
1Lots of times in analysis, integrals are defined via Darboux integrals or the equivalent Riemann integral which can be found here https://en.wikipedia.org/wiki/Riemann_integral, https://en.wikipedia.org/wiki/Darboux_integral. However as mentioned by @Qiaochu Yuan, integrals can be defined in the context of measure theory as well, which is where my knowledge ceases. – Anthony Corsi Mar 27 '25 at 22:11
-
2This question and others linked by stackexchange may answer your question: Why can't the second fundamental theorem of calculus be proved in just two lines?. If you believe none does, please [edit] the question, make it clear how it’s different and/or how the answers on those questions are not helpful for your problem.s – Ethan Bolker Mar 27 '25 at 23:42
-
2Don't overthink traditional notation. It communicates ideas, can be very suggestive and useful, and those are reasons it's stuck around so long, but it's a convention. When you take integration theory you'll see things like $\int f$. – A rural reader Mar 27 '25 at 23:45
2 Answers
I prefer the answer that $dx$ is a meaningless symbol which produces for you the correct answer when you use substitution. For example, if you want to integrate $\int_0^1 e^{x^2} ~ 2x ~ dx$ then the substitution $y = x^2$ will result in $dy = 2x ~ dx$ and so the integral transforms $\int_0^1 e^y ~ dy$.
Now, $dx$ and $dy$ are meaningless symbols on their own, and they cannot be divided, but if you use this "symbolic non-sensical notation", then it happens to give you the correct answer.
Of course, one can be more sophisticated and try to give them meaning, but in actuality most people do not assign meaning to them and use them as symbols that carry out the integration for you.
This is my preferred answer because all the "fancy" answers are of no use to an introductory student who is first learning calculus. By simply treating these symbols are mnenomics that carry out substitution for you, you begin to understand what they are about. But if you try to understand the "fancy" answers then you just end up confusing yourself.
The most ironic answer to your question, "what do they mean?", is simply "they do not mean anything", and the irony is that by not giving an answer, explains more (at least to a first time student) than any other sophisticated explanation that people give.
- 5,553
-
Thank you for the explanation, and I understand your perspective. However, I have acted as if they mean nothing ever since taking calculus at the very beginning of college, but I am now a second year and learning more about math, and I think I am no longer satisfied with that explanation, even if going further is very complicated. I am simply too intrigued. – Ryder Mendelson Mar 28 '25 at 09:15
With Riemann integration, $$\int_a^b f(x)\,dx=\lim\sum f(x^*_i)\Delta x_i$$ provided that limit exists. The sum is over a partition of $[a,b]$, with each part's length being $\Delta x_i$. And $x^*_i$ is just some number in the $i$th part. The limit is over some sequence of partitions such that the largest part's length approaches zero. This is only well defined if the limit is the same regardless of what that sequence of partitions actually is and regardless of what numbers $x^*_i$ are used.
So here with a definite integral, the $dx$ is a symbolic reminder of the $\Delta x_i$. It is a symbolic reminder that all the $\Delta x_i$ were getting smaller (on the whole) approaching zero. It's at least consistent with viewing $dx$ as an infinitesimal width.
- 56,991