Electric current notation

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 $\vec{F} = \frac{d\vec{p}}{dt}$ ?

• 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. – ACuriousMind May 1 '15 at 14:20
• 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? – mwa1 May 1 '15 at 14:56
• 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? – ACuriousMind May 1 '15 at 15:11
• "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 $\delta t$). – mwa1 May 1 '15 at 15:50
• For more on the differences of derivatives/differentials, see this Physics.SE post – Kyle Kanos May 2 '15 at 12:14