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

mwa1
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    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 01 '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 01 '15 at 14:56
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    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 01 '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 01 '15 at 15:50
  • Are you sure it's $\delta q$ and not $\partial q$? – Demosthene May 01 '15 at 15:54
  • Absolutely, it reads $I = \frac{\delta q}{\delta t}$ and is defined as " the quantity of charge $\delta q $ going through the section of a conductor between $t$ and $t+\delta t$ " – mwa1 May 01 '15 at 16:03
  • Then I'd say the following: $\frac{dq}{dt}$ is the total derivative of $q(t)$, $\frac{\partial q}{\partial t}$ is the partial derivative of $q(t,x_1,x_2,\ldots)$ (some function of $t$ and something else); $\Delta q$ is a large/macroscopic variation of $q$, whereas $dq$ is its infinitesimal counterpart, and finally $\delta q$ is somewhere in between, i.e. it is not necessarily infinitesimal. You'll find this $\delta$ symbol frequently in the calculus of variation - $\delta q$ is precisely a variation in $q$. – Demosthene May 01 '15 at 16:54
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    For more on the differences of derivatives/differentials, see this Physics.SE post – Kyle Kanos May 02 '15 at 12:14
  • $\uparrow$ Which sources? – Qmechanic May 02 '15 at 16:42
  • "Which sources" : different textbooks, the internet, teachers – mwa1 May 03 '15 at 18:02

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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!

slhulk
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  • 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... – mwa1 May 01 '15 at 17:23