Are the symbols $d$ and $\delta$ equivalent in expressions like $dy/dx$? Or do they mean something different?
Thanks
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8$\begingroup$ 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". $\endgroup$– Giuseppe NegroFeb 28, 2013 at 23:41
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2$\begingroup$ $\partial$ is a modified form of $\delta$, but is not the same thing. $\endgroup$– George V. WilliamsFeb 28, 2013 at 23:43
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$\begingroup$ I don't think is accurate to call $\partial$ and $\delta$ as a modified versions of each other. they look like mirror images of each other, but a $\partial$ isn't the greek letter delta letter... i think $\partial$ is a modified letter "d"... but then again..who knows because the greek letter $\delta$ is the lowercase letter "d" in the latin alphabet... $\endgroup$– Bill MooreDec 21, 2019 at 21:21
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$\begingroup$ So should the question be edited to contain ∂ instead of δ? $\endgroup$– ArchLinuxTuxJun 23, 2020 at 16:29
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4 Answers
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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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4$\begingroup$ 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. $\endgroup$ Mar 1, 2013 at 0:18
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6$\begingroup$ 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. $\endgroup$– isarandiOct 13, 2014 at 14:17
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1$\begingroup$ 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 $\endgroup$– SteevenDec 17, 2014 at 17:06
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2$\begingroup$ @isarandi Most of the time we could use partial derivatives in place of regular derivatives and nothing would change. The only exception I can think of is if we have $f(t,x)$ a function of two variables, and $x$ is a function of $t$. Then $df/dt=\partial f/\partial t+\partial f/\partial x dx/dt$, so that $df/dt$ and $\partial f/\partial t$ are two different things. $\endgroup$– TeepeemmFeb 26, 2018 at 15:48
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2$\begingroup$ @Teepeemm True. But despite its popularity I find that notation somewhat sloppy. If you want the d version than you can just say $\partial f(t, x(t)) / \partial t$ (using $\partial$ as a universal differentiation sign, since the distinction is only relevant if you use implicit notation, where it's not clear if something is a function or a variable). $\endgroup$– isarandiFeb 28, 2018 at 12:47
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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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1$\begingroup$ Nope. $\partial$ is used for partial derivatives. $\endgroup$ May 31, 2020 at 10:20
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Here's the dummy version:
∂ is used when a function; say f(x,y,z) depends on more than 1 variable.
d is used when a function; say f(t) depends on only 1 variable.
Example:
You have f(x,y) and x(t) and y(t)
In the end f only depends on t here, but in the chain it can depend on x or y, so you write:
df/dt = ∂f/∂x * dx/dt + ∂f/∂y * dy/dt
You use df/dt, dx/dt and dy/dt here because they only depend on t. You use ∂f/∂x and ∂f/∂y because for f in this partial differential step f can go via both x or y.
I hope this helps.
Good Luck
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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.
