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Physicists often deal with equations like $dW = dq(V_a - V_b)$ and say "Don't tell this your math teacher" when they do $\frac {dW}{dt} = \frac {dq}{dt}(V_a - V_b)$. But why is it so dangerous to differentiate against time?

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    It's not. It's dangerous to "divide" by $dt$. – Arthur Sep 06 '17 at 20:35
  • @Arthur Unless you are Abraham Robinson. Then it's okay. :) – Xander Henderson Sep 06 '17 at 20:36
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    Pesky side note: it is differentiate not derive. – Mosquite Sep 06 '17 at 20:36
  • They are often dealing with multivariate functions and the $dy$ actually refers to an infinitesimal change $\delta y$. Only later is the "derivative" actually taken, if one does exist. – rivendell Sep 06 '17 at 20:38
  • No one asked me, but it has always scared me how physicists use differentials. I mean, I'm sure it's fine, but... – Randall Sep 06 '17 at 20:39
  • @Randall I don't know---I kind of like the notation $\int \mathrm{d}x f(x)$ for the integral of $f$. It treats $\int\mathrm{d}x$ as the operator that it always wanted to be. – Xander Henderson Sep 06 '17 at 20:46
  • That doesn't bother me. – Randall Sep 06 '17 at 20:48
  • @Randall Neither me. The plasticity of Leibniz notations (dx, dy,dx/dy...) explains the huge progresses accomplished in the18th century, either on the mathematics side or on the physics side. This is still true, even in mathematical research, where "infinitesimal thinking" in the 18th century style, can help a lot. Look at the writings of Poincaré, one of the last to have been a great mathematician and a great physicist: he uses differentials, for example for building modern thermodynamics theory, in a very intuistic manner. – Jean Marie Sep 06 '17 at 21:18
  • @JeanMarie I would also suggest looking at Abraham Robinson's non-standard analysis. It is a much more recent (and quite successful) attempt at making the 17th century intuition rigorous to a 20th century standard. With a bit of care, infinitesimal thinking can be made quite rigorous. – Xander Henderson Sep 07 '17 at 02:43
  • @Xander Henderson You are right, I should have mentionned it. – Jean Marie Sep 07 '17 at 04:29

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