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As an engineer, I am a bit confused about the interpretation of differential. Sometimes, they are used as infinitesimal quantities that can be treated as factors, e. g. $$\frac{dy}{dx}=1\space\vert\cdot dx$$ $$dy=dx\space\vert\int$$ $$y=x+C$$ whereas in other contexts, they are treated as operators, e. g. $$f(x,y)=f(u(x,y),v(x,y))\space\vert\frac{\delta}{\delta x}$$ $$\frac{\delta f}{\delta x}=\frac{\delta f}{\delta u}\cdot\frac{\delta u}{\delta x}+\frac{\delta f}{\delta v}\cdot\frac{\delta v}{\delta x}$$ where treating the differentials as factors would lead to the reduced expression $$\frac{\delta f}{\delta x}=\frac{\delta f}{\delta x}+\frac{\delta f}{\delta x}=2\cdot\frac{\delta f}{\delta x}$$ and ultimately $$1=2$$ which is obviously false. So what am I missing? Does it have something to do with the notation ($dx$ vs $\delta x$)?

chicken_game
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  • At the very least, partial differentials ($\partial$, not $\delta$) are very different from total differentials ($d$, though some insist on $\mathrm d$). Ultimately, though, in regular calculus, something like $\frac{df}{dx}$ is a single symbol that represents a limit, and you can't really split it up and give meaning to each part (except arguably the $\frac{d}{dx}$ and the $f$). But if you go beyond calculus, there are ways to do it. – Arthur Dec 16 '20 at 11:27
  • So what allows us to split the differential in my first example? – chicken_game Dec 16 '20 at 11:58
  • Luck. The limits are nice enough that it just happens to work out. – Arthur Dec 16 '20 at 12:41
  • What do you mean by limits in this context? – chicken_game Dec 16 '20 at 12:52
  • The definition of the derivative is a limit. That's hidden away when we write $\frac{df}{dx}$, but it is most definitely there. And it dictates what kind of manipulations are valid, and why. – Arthur Dec 16 '20 at 13:10
  • Ok, so from what I've read I think it's fair to say that $\frac{df}{dx}$ is not a fraction, but merely Leibniz's notation for the limit resulting in the first derivate (which is more obvious for higher order derivatives which are written as $\frac{d^nf}{dx^n}$. Yet in some cases, as you said, it is permissible to treat them as fractions, which I've often seen being done. Is there a precise explanation as to when this is possible? (After all, relying on "luck" seems like a risky strategy in maths ;) – chicken_game Dec 16 '20 at 15:20
  • I found an explanation in one of the related questions that seems fancy to me (can't find the link atm): It said we can treat differentials as factors as long as we remember the conditions they depend on. In my first example above, we can set $dy=dx$ because we know that the "dependent differential" (i. e. dependent on the size of $dx$) $dy$ increases at the same rate as the "independent differential" $dx$, whereas in my second example, $\delta u$ and $\delta v$ each appear as inedependent (denominator) and dependent differentials (enumerator), which is why they don't cancel out. Fair enough? – chicken_game Dec 16 '20 at 15:44

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