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Throughout undergraduate physics textbooks, you will see informal math with differentials where elements like $dx$ and $dy$ are multiplied around like scalar constants, and differentiation in terms of a variable is treated as analogous to division. What is the theoretical justification for this? I have never seen a formal mathematical argument to say why this can be done, especially not in the textbooks that use it. When I mean formal, I mean an argument from the point of view of rigorous mathematics, not just saying that $\Delta x/\Delta y$ approximates $dx/dy$ so we can treat $dx$ like we would $\Delta x$. Are there any formal proofs available?

An example of the type of differential mathematics I am talking about is used in thermodynamics. https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation I have never seen the formal justification that undergirds this way of talking about infinitesimal changes and using the differentials like constants.

sakurashinken
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    Physics may get sloppy with the math at times, but it works so we put up with it. –  Sep 23 '20 at 17:58
  • The formal proofs involve taking limits and working with big-O notation. They can be done, so we don't do it because we know we can. – Javier Sep 23 '20 at 18:00
  • Would any one be willing to demonstrate such a proof or direct me to where I can find one? – sakurashinken Sep 23 '20 at 18:01
  • Golden rule of physicists: (Most) physicists know what is allowed to do and what leads to sound results even if the mathematics used at first sight looks scary. In particular in case of differentials, learn about differential forms which give symbols like $dx$ and $dy$ a sound mathematical definition. – Frederic Thomas Sep 23 '20 at 18:16
  • Can you explain in more detail what you are looking for? I have even seen in mathematics text books that we can solve the differential equation $$\frac{\text d y}{\text dx}=f(y)$$ as $$\int\frac{\text dy}{f(y)}=\int\text dx$$ Why isn't dealing with $\text dx$ as $\Delta x\to0$ sufficient? These will help narrow down what you are looking for. – BioPhysicist Sep 23 '20 at 18:24

2 Answers2

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There is no known logically rigorous justification that works in all instances. But it is immensely useful as a heuristic, and it focuses attention where it needs to be, and it keeps things dimensionally correct (e.g. if $f(x)$ is in meters per second and $dx$ is in seconds, then $f(x)\,dx$ is in meters, and if $s$ is in meters and $t$ in seconds, then $d^2 s/dt^2$ is in meters per second per second, etc.).

Michael Hardy
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    Could you suggest what justification covers the most common physics uses and, if so, can you outline that justification ? – StephenG - Help Ukraine Sep 23 '20 at 20:20
  • That would be much appreciated. I feel like science is full of unjustified truisms like this that everyone just repeats without knowing where they came from and why. – sakurashinken Sep 23 '20 at 20:45
  • @sakurashinken : I think it's crystal-clear where they came from, just based on what you see in physics courses. $\qquad$ – Michael Hardy Sep 23 '20 at 21:00
  • @sakurashinken : I wonder if the question you asked differs from the one whose answer would satisfy you. Maybe I'll be back$,\ldots \qquad$ – Michael Hardy Sep 23 '20 at 21:24
  • I am wondering about the formal proof that would justify the heuristic clarity. We have definitions for the limit and for Gausian integers that are meant to satisfy this type of seeming mysticism. If it is taken as a definition, that would be fine for me. It doesn't seem that way though, and seems like it can be broken down into deeper principles. Perhaps I'll have to research myself. – sakurashinken Sep 23 '20 at 22:50
  • @sakurashinken : What your actual question is is unclear to me. Certainly calculus has been logically rigorously justified. – Michael Hardy Sep 24 '20 at 00:43
  • @sakurashinken : And what do Gaussian integers have to do with it? – Michael Hardy Sep 24 '20 at 00:43
  • $$ \begin{align} y & = x^5 \ {} \ \left( \frac{dy}{dx} \right. & = \left. \lim_{\Delta x,\to,0} \frac{\Delta y}{\Delta x} \right) = \lim_{\Delta x,\to,0} \frac{(x+\Delta x)^5 - x^5}{\Delta x} \ {} \ & = \lim_{\Delta x,\to,0} \frac{(\Delta x)\Big( (x+\Delta x)^4 + (x+\Delta x)^3 x + (x+\Delta x)^2 x^2 + (x+\Delta x) x^3 + x^4 \Big)}{\Delta x} \ {} \ & = \lim_{\Delta x,\to,0} \Big( (x+\Delta x)^4 + (x+\Delta x)^3 x + (x+\Delta x)^2 x^2 + (x+\Delta x) x^3 + x^4 \Big) \ {} \ & = x^4 + x^4 + x^4 + x^4 + x^4 = 5x^4. \end{align} $$ – Michael Hardy Sep 24 '20 at 00:57
  • $\qquad \uparrow \quad$ @sakurashinken$\qquad \uparrow \quad$ $$ \left( \frac{dy}{dx} = \lim_{\Delta x,\to,0} \frac{\Delta y}{\Delta x} \right) $$Are you familiar with things like$\quad\uparrow\quad$this (above)? $\qquad$ – Michael Hardy Sep 24 '20 at 00:58
  • What I mean is that mysterious notation usually has underlying logical justification, and I've never seen it formally presented for differential arithmetic. In the case of limits we have the delta epsilon definition and for complex numbers we have definitions using tuples with special operations for addition and multiplication. A great example of the type of math I am enquiring about is en.wikipedia.org/wiki/Fundamental_thermodynamic_relation. What is the formal justification for this type of notation? I've never seen it and don't think it is often presented. – sakurashinken Sep 24 '20 at 03:56
  • @sakurashinken : Your Question has no concrete example or examples to provide traction for an answer. Since you don't give an example, it is impossible to rigorously explain what is meant by the physicists' manipulations. I might guess what you are talking about, but your comments here make it clear that at least one guess by at least one answerer does not respond to whatever example you have not provided. – Eric Towers Sep 24 '20 at 05:48
  • @sakurashinken : The example that is hidden here in comments and does not appear in the Question, which omission you very much should correct by editing it into your Question, is a total differential equation. It's implicit differentiation with respect to an anonymous variable. The division you ask about in your Question is a rearrangement of the chain rule. (continued) – Eric Towers Sep 24 '20 at 06:00
  • @sakurashinken : (...) For $a$ the anonymous variable, $$\frac{\mathrm{d}U}{\mathrm{d}a} = T \frac{\mathrm{d}S}{\mathrm{d}a} - P \frac{\mathrm{d}V}{\mathrm{d}a} \text{.}$$ Now divide through by $\frac{\mathrm{d}S}{\mathrm{d}a}$ and use the chain rule in the form $$\frac{\frac{\mathrm{d}U}{\mathrm{d}a}}{\frac{\mathrm{d}S}{\mathrm{d}a}} = \frac{\mathrm{d}U}{\mathrm{d}S}\text{.}$$ – Eric Towers Sep 24 '20 at 06:00
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Although the question is strictly mathematical, there is an important physical/philosophical point to be made which hasn't yet been mentioned, especially since you originally asked this question on the physics website. You say that physical models approximate mathematical models, but it is arguably the other way round! We need to remember that our models are only as valid as our experimental abilities and experimentally continuity doesn't exist: you can measure something very, very, very precisely but it will never be a real number. So one reason physicists are "sloppy" is very straightforward: if you need to model something that you know exists and is finite, you don't need to prove existence theorems, and you also don't need to prove that it converges, which a lot of mathematics is concerned with.

Another reason why physicists use these mathematical idealisations is because they are much more convenient (discrete maths is much harder to manipulate than continuous maths and is also less developed on the whole). Many people like to gloss over this point, but we also need to remember that a lot of rigorous mathematics has been largely inspired by the approximate nature of physics (e.g. distribution theory, calculus, functional analysis, etc) and there are still some concepts that work but are not considered rigorous, such as real-time path integrals, meaning that just because something hasn't yet been proven, it might still be physically useful, relevant and experimentally validated - and I would argue this, rather than mathematical purity, is the main purpose of theoretical physics.

Godzilla
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