Difference between $\Delta$, $d$ and $\delta$

Question

I have read the thread regarding 'the difference between the operators $\delta$ and $d$', but it does not answer my question.

I am confused about the notation for change in Physics. In Mathematics, $\delta$ and $\Delta$ essentially refer to the same thing, i.e., change. This means that $\Delta x = x_1 - x_2 = \delta x$. The difference between $\delta$ and $d$ is also clear and distinct in differential calculus. We know that $\frac{dy}{dx}$ is always an operator and not a fraction, whereas $\frac{\delta y}{\delta x}$ is an infinitesimal change.

In Physics, however, the distinction is not as clear. Can anyone offer a clearer picture?

Answer 1

The symbol $\Delta$ refers to a finite variation or change of a quantity – by finite, I mean one that is not infinitely small.

The symbols $\mathrm{d}$ and $\delta$ refer to infinitesimal variations of numerators and denominators of derivatives.

The difference between $\mathrm{d}$ and $\delta$ is that $\mathrm{d}X$ is only used if $X$ without the $\mathrm{d}$ is an actual quantity that may be measured (i.e. as a function of time) without any ambiguity about the "additive shift" (i.e. about the question which level is declared to be $X=0$). On the other hand, we sometimes talk about small contributions to laws that can't be extracted from a well-defined quantity that depends on time.

An example, the first law of thermodynamics. $$\mathrm{d}U = \delta Q - \delta W$$ The left hand side has $\mathrm{d}U$, the change of the total energy $U$ of the system that is actually a well-defined function of time. The law says that it is equal to the infinitesimal heat $\delta Q$ supplied to the system during the change minus the infinitesimal work $\delta W$ done by the system. All three terms are equally infinitesimal but there is nothing such as "overall heat" $Q$ or "overall work" $W$ that could be traced – we only determine the changes (flows, doing work) of these things.

Also, one must understand the symbol $\partial$ for partial derivatives – derivatives of functions of many variables for which the remaining variables are kept fixed, e.g. $\partial f(x,y)/\partial x$ and similarly $y$ in the denominator.

Independently of that, $\delta$ is sometimes used in the functional calculus for functionals – functions that depend on whole functions (i.e. infinitely many variables). In this context, $\delta$ generalizes $\mathrm{d}$ and has a different meaning, closer to $d$, than $\delta$ in the example of $\delta W$ and $\delta Q$ above. Just like we have $\mathrm{d}y=f'(x)\mathrm{d}x$ for ordinary derivatives in the case of one variable, we may have $\delta S = \int_a^b \mathrm{d}t\,C(t)\delta x(t)$ where the integral is there because $S$ depends on uncountably many variables $x(t)$, one variable for each value of $t$.

In physics, one must be ready that $\mathrm{d}$, $\delta$, $\Delta$ may be used for many other things. For example, there is a $\delta$-function (a distribution that is only non-vanishing for $x=0$) and its infinite-dimensional, functional generalization is called $\Delta[f(x)]$. That's a functional that is only nonzero for $f(x)=0$ for every $x$ and the integral $\int {\mathcal D}f(x) \,\Delta[f(x)]=1$. Note that for functional integrals (over the infinite-dimensional spaces of functions), the integration measure is denoted ${\mathcal D}$ and not $\mathrm{d}$.

Answer 2

In many books, the difference between $d$ and $\delta$ is that, in the first case, we have the differential of a function and, in the second case, we have the variation of a functional.