I have read the few questions on calculi notation, particularly the notations on partial and total derivatives. My question seems to have not been answered, or at least not brought to my attention. If there has been a similar thread, please direct it to me. Thanks :)

Anyways, my question is simple: what's the difference between $\frac{\delta}{dt}$ and $\frac{d}{dt}$? In other words:

$$ \frac{\delta x}{\delta y} \text{ is } \frac{dx}{dy} \text{?} $$

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    $\begingroup$ Are you sure you mean $$\frac{\delta}{dt}$$ and not $$\quad \frac{\partial}{\partial t}\quad ?$$ $\endgroup$ – Zev Chonoles May 19 '13 at 4:32
  • $\begingroup$ or $\delta/\delta t$? $\delta/ dt$ seems rather strange. $\endgroup$ – Brady Trainor May 19 '13 at 4:34
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    $\begingroup$ @Brady: I've never seen $\delta$ used for a derivative notation. Note that $\delta$ (\delta) and $\partial$ (\partial) are different symbols. $\endgroup$ – Zev Chonoles May 19 '13 at 4:36
  • $\begingroup$ @ZevChonoles, I understand, but I do have a vague recollection of seeing such notation, though it must be rare. Take for instance the divergence of notations in physics treatments, for instance $\int dx f$. $\endgroup$ – Brady Trainor May 19 '13 at 4:38
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    $\begingroup$ Notations like $\delta L/\delta f$ are commonly used in the calculus of variations to denote the derivative of a functional (i.e., a function whose domain is a space of functions) with respect to the unknown function. $\endgroup$ – Jack Lee May 19 '13 at 4:42

We have a very good explanation on the $\delta$ notation in the Chapter 6 of the book "Classical Dynamics" by Thornton and Marion. Despite the fact that it is a book for physicists you will see the explanation of such a notation by means of the calculus of variations. It's difference from the other simbols for derivatives is also covered there.

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