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Often I find the following expressions in physics books: Say we have a current density $\vec{j}=\rho\vec{v}$ through a surface $\vec{F}$ of particles $N$ in the volume $V$ with the density $\rho=dN/dV$ and -for simplicity- the velocity $\vec{v}\,||\,d\vec{F}$, $\vec{v}=d\vec{s}/dt$. Now one writes $$ \rho\vec{v}=\frac{dN}{dV}\frac{d\vec{s}}{dt}=\frac{dN}{dsdF}\frac{d\vec{s}}{dt}=\frac{dN}{dFdt}\hat{s}$$ with $$ \vec{j}\cdot\vec{dF}=\frac{dN}{dt}$$. My question: Shouldn't one rather write $\frac{d^2N}{dFdt}\hat{s}$ since $N(V,t)$? The same appears in expressions like $$\vec{\nabla}\cdot\vec{j}=-\frac{1}{dV}\frac{dN}{dt}=-\frac{1}{dt}\frac{dN}{dV}=-\frac{d\rho}{dt}$$ which is the continuity equation. Of course in the latter $N$ is infinitesimally small quantity since $N(dV)$. From the physical point of view it makes perfectly sense but from the mathematical form it looks inconsistent. How to cure it?

pawel_winzig
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  • $F$ is a 2-D surface, so by your logic you could equally well say that it should be $d^3 N/dx dy dt$ or something like that instead. (Not that I'm saying that you should say that, mind you.) – Michael Seifert Jun 16 '16 at 13:36
  • See http://physics.stackexchange.com/questions/154373/physics-derivatives-written-in-a-weird-way?rq=1 – Virgo Jun 16 '16 at 18:07
  • @Virgo: I have no problem with differentials, I have a problem with the inconsistent treatment of them in physics as shown above. – pawel_winzig Jun 17 '16 at 07:13
  • As mentioned by a commentator in that link, in physics differentials are usually treated as small perturbations. This doesn't cause any problems, in general, due to the well behaved nature of the functions considered. – Virgo Jun 17 '16 at 17:55
  • @Virgo: Sounds like a cheap excuse. One part of my question was: Shouldn't it be $d^2N/(dFdt)$ . This concerns also "well behaved" functions. The same holds for this strange looking expression $1/dV\cdot dN/dt$: The limit $V \rightarrow 0$ cannot be taken independently for $(dN(V)/dt)$ and $1/\Delta V$ but it has the appearance. – pawel_winzig Jun 19 '16 at 12:17
  • I think this is really just a matter of taste. Most notation need to omit some information and so one has to rely on the reader keeping in mind certain facts which aren't explicitly stated. – Virgo Jun 19 '16 at 17:51

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