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Depending on the source, I sometimes read $\frac{\delta q}{dt}$ , $\frac{dq}{dt}$ or even $\frac{\delta q}{\delta t}$ (rare)

Wich one is the correct notation ?

In theory we are to know if a differential form is exact before we can write $dq$ or $dt$, but how are we supposed to do that ?

Physics books usually choose a notation without giving much explanations... (actually I've only seen explanations about this in Thermodynamics, for $\delta Q$ and $\delta W$)

What tells me for sure that I can write $\vec{F} = \frac{d\vec{p}}{dt}$ ?

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    $\begingroup$ Uh...you can write $\mathrm{d}\omega$ for any differential form $\omega$. Exactness would mean $\mathrm{d}\eta = \omega$ for some $\eta$. I don't understand your question. $\endgroup$ – ACuriousMind May 1 '15 at 14:20
  • $\begingroup$ I'm not sure I understand it myself, I find this confusing. As I understand it, I can only write $dq = I dt$ if $I = \frac{dq}{dt}$. But how do I know this ? is it only by definition ?(it looks like a chicken and egg problem). Then why can't I write $F = \frac{dW}{dx}$ for the work of a force? $\endgroup$ – mwa1 May 1 '15 at 14:56
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    $\begingroup$ Work is defined as a line integral $W := \int_\gamma F$, this only is the same as $F = \mathrm{d}W/\mathrm{d}x$ if the line integral does not depend on the path taken, but onyl on the endpoints. Current, on the other hand, is defined as $I := \mathrm{d}q/\mathrm{d}t$. What is confusing about that? $\endgroup$ – ACuriousMind May 1 '15 at 15:11
  • $\begingroup$ "What is confusing about that" : Mostly the inconsistency of notations across different sources. As I said, I just saw the current defined as $I = \frac{\delta q}{\delta t}$ (and it was the first time I had ever seen $\delta t$). $\endgroup$ – mwa1 May 1 '15 at 15:50
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    $\begingroup$ For more on the differences of derivatives/differentials, see this Physics.SE post $\endgroup$ – Kyle Kanos May 2 '15 at 12:14
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The notation whether it be d or delta doesn't matter as long as it describes an element (a minute amout) of the quantity.

Please keep in mind that this is NOT a ratio. So you can't write

dq = I. dt

This is mathematically wrong. As differentiation is an operation and not a mere ratio. It is like a machine and you can't separate it's parts or the machine won't work!

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  • $\begingroup$ Yes, I remember having read that before. And I think that is another reason why there is so much confusion about the notations, because we can see this in every undergraduate physics book. It looks like a ratio and behaves as such, but it's not... $\endgroup$ – mwa1 May 1 '15 at 17:23

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