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I've seen people write integrals like $$\int{\sqrt{dx}}$$, where the differential is not just appended to the end of the integrand in the way I'm used to, but is being acted on by some function in the scope of the integral (at least, that's what it seems like to me). How should I interpret this kind of notation?

Rando McRandom
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  • where have you seen this? – user619894 Jul 19 '23 at 18:37
  • @user619894 In this introduction to calculus of variations, where they write an integral which is supposed to be the total length of a curve between two points A and B. I've seen things like it elsewhere as well, but I can't recall those examples right now. – Rando McRandom Jul 19 '23 at 18:45
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    in fact the expression was $\sqrt{dx^2+dy^2}$ which is dimensionally correct – user619894 Jul 19 '23 at 19:11
  • @user619894 Can you (or someone else) please explain (or to point some explanation) of how I'm supposed to understand an expression like that? I'm only used to seeing integrals of the form $$\int{f(x)}dx$$. – Rando McRandom Jul 19 '23 at 19:31
  • It will be helpful, and possibly duplicate: [Does an integral of the form $\int f(x) , \sqrt{dx}$ have any meaning?] - https://math.stackexchange.com/questions/3298827/does-an-integral-of-the-form-int-fx-sqrtdx-have-any-meaning –  Jul 19 '23 at 19:35
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    As user619894 has pointed out, you did not see "$\sqrt dx$". You saw "$\sqrt{dx^2 + dy^2}$", which is a much different beast. If you have a curve $\gamma(t) = (x(t), y(t)), a\le t \le b$, then $\int_\gamma \sqrt{dx^2 + dy^2}$ is defined by $$\int_\gamma \sqrt{dx^2 + dy^2} = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2},dt$$ which makes notational sense if you pretend the various differentials act like numbers. But there is no such notational sense to be made of $\int \sqrt{dx}$ There are other cases where $\sqrt {dW}$ makes sense, but they are esoteric. – Paul Sinclair Jul 20 '23 at 16:28
  • @PaulSinclair Thank you. That's very helpful. I would be grateful if someone could point me to resources for understanding how to use/understand differentials as numbers. For some reason this wasn't coverd in the calculus books I read. – Rando McRandom Jul 21 '23 at 08:56
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    I sincerely doubt that. I have never seen an introductory calculus book that didn't cover it. You just didn't understand what the books were telling you, because there were too many other concepts that you were still coming to grips with at the time. Informally, Call a product of differentials and real numbers to be of degree $k$ if there are $k$ differentials in the multiplication. The only difference in behavior between differentials and numbers is that when summing terms involving differentials, only the terms of lowest degree contribute to the sum. – Paul Sinclair Jul 21 '23 at 11:38

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