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Are the symbols $d$ and $\delta$ equivalent in expressions like $dy/dx$? Or do they mean something different?

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As far as I can see, the symbol $\delta$ is not used to mean differentiation in pure mathematics. On the contrary, in physics it is sometimes used instead of $d$ to signal that there is something to be aware of. For example, you sometimes used $\delta$ instead of $d$ to mean "functional derivative". –  Giuseppe Negro Feb 28 '13 at 23:41
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$\partial$ is a modified form of $\delta$, but is not the same thing. –  George V. Williams Feb 28 '13 at 23:43

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As Giuseppe Negro said in a comment, $\delta$ is never used in mathematics in $$\frac{dy}{dx}.$$

(I am a physics ignoramus, so I do not know whether it is used in that context in physics, or what it might mean if it is.)

You do sometimes see $$\frac{\partial y}{\partial x}$$

which means that $y$ is a function of several variables, including $x$, and you are taking the partial derivative of $y$ with respect to $x$. This is a slightly different meaning than just $\frac{dy}{dx}$. For example, suppose that $f(x,y)$ is a function of both $x$ and $y$, and that each of $x$ and $y$ can in turn be expressed as functions of a third variable, $t$. Then one can write:

$$\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$

The $\partial$ symbol is not a Greek delta ($\delta$), but a variant on the Latin letter 'd'. In $\TeX$, you get it by writing \partial.

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Might I add that in continuum mechanics the distinction between $d/dt$ and $\partial/\partial t$ is relevant? Indeed, the first stands for "material derivative" while the second is an ordinary partial derivative. –  Giuseppe Negro Mar 1 '13 at 0:18
    
But a single variable function is just a special case of a function taking n variables. So can we use partial derivatives in the case where we have only 1 input variable to the function? It's still a partial derivative in a general sense. For me it would be more consistent if we got rid of the exceptional notation for the case of 1 variable. –  isarandi Oct 13 at 14:17
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Is $\delta$ really never used in math as the derivative symbol? E.g. it is used as example for a functional derivative here: en.wikipedia.org/wiki/Functional_derivative –  Steeven 2 days ago

The Greek letter $\delta$ is never used in correctly typeset differential quotients. You probably saw the \partial symbol, e.g. $\dfrac{\partial f(x,y)}{\partial x}$ which here is used to denote the partial derivative of $f$ with respect to its first variable.

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I am so excited that I can help! I just learned about this in Thermodynamics (pg 95 of Fundamentals of Thermodynamics, Borgnakke & Sonntag). ' "d" stands for the exact differential (as often used in mathmatics); where the change in volume depends only on the initial and final states. "δ" refers to an inexact differential (which is used in physics when calculating things like work), where the quasi-equilibrium process between the two given states depends on the path followed. The differentials of path functions are inexact differentials and designated by, "δ." '

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