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I know this has been covered before in this post, but I still don't fully understand why cancelling differentials —not just inside an integral, but in general— is not considered okay or valid among mathematicians.

I'm currently studying physics at college, and my teachers —those who are physicists and not mathematicians, at least— do it all the time without it causing any trouble. In fact, the first (and less rigorous) method we were taught for solving differential equations relies heavily on treating differentials like numbers and derivatives like fractions. I don't know if it has a proper name, but I mean something like:

$$ \frac{\text{d}I}{\text{d}t}\cos(\omega t) = I\omega \sin(\omega t) \implies \frac{\text{d}I}{I}=\omega \tan(\omega t)\space\text{d}t $$

Now, I know that differentials aren't 'really' numbers. I mean, they can be understood both as the "linear part" of a variation or as a linear form, and the former looks like it should be okay to treat them like numbers.

Even the chain rule and the derivative of inverse functions seem to tell us that treating them like so should be consistent, when you write them using Leibniz's notation.

Is there really a situation where treating differentials like numerators and denominators causes any trouble?

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  • $\begingroup$ There is. You can find some examples in a good calculus book. $\endgroup$
    – Jakobian
    Commented Jun 18, 2018 at 18:31
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    $\begingroup$ Related questions: (1), (2), (3) $\endgroup$
    – JMoravitz
    Commented Jun 18, 2018 at 18:32
  • $\begingroup$ Regarding your example with separable differential equations, see this question: math.stackexchange.com/questions/27425/… $\endgroup$ Commented Jun 18, 2018 at 18:39
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    $\begingroup$ I remember several questions on this site where students try to extend this method of solving an ODE to second order equations and we end up with $y'' = e^x$ becoming $\iint d^2 y = \int e^x(dx)^2$ whatever that means. $\endgroup$
    – Winther
    Commented Jun 18, 2018 at 18:40
  • $\begingroup$ @Adam could you name one, please? $\endgroup$
    – TeicDaun
    Commented Jun 18, 2018 at 18:50

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