Delta operator magic

What does it mean when, in (derivation of) (partial) differential equations, a term containing an operator like:

$\delta t$

gets replaced by:

$d t$

and what are the rules pertaining to this? What is the difference between the two? It seems black magic to me :).

Thanks!

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There are several ways to formalize this. One is that we're just taking a limit as delta t becomes small, so difference quotients become derivatives. In this formalization "dt" is not a number, it's part of a symbol like df/dt for differentiation. – Qiaochu Yuan Feb 1 '11 at 16:06

Could you give a specific example? In some instances $\delta t$ is used to represent a variation or change in time, in this case. $dt$ just represents the "differential" of time in this case, i.e., the infinitesimal change. Think of it as the differential in calculus, which is obtained by passing to the limit in the expression $$\frac{\Delta y}{\Delta x}$$ yielding $$\frac{d y}{d x}.$$
Incidentally, $\delta t$ is also very used in the calculus of variations, but its interpretation there is somewhat different. Compare en.wikipedia.org/wiki/Differential_(infinitesimal), with en.wikipedia.org/wiki/First_variation – Jose L. Lykón Feb 1 '11 at 16:14