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I recently stumbled upon the notation $d^2N = dn\,dV$ in physics classes. Someone told me that it's a second-order differential element because it's a product of two differential elements, hence the notation $d^2$

Is this related to the notation used for second order derivatives $\frac{d^2f}{dx^2}$ ?

Also, I'm confused about differentials properties:

if $z = xy$

$dz = d(xy) = x \, dy + y \,dx$

but $d^2z = dx \, dy$

How does it work, i'm lost...

mvw
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user1234161
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1 Answers1

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No, it is not related. The first is a volume, or rather surface element, so the exponent $2$ denotes the geometrical dimension. For the second order example the $2$ means a two times application of a differentiation operation.

About the properties: The first one is an example of the differentiation rules. The second one, again, is preparing a volume or surface element for an integration of some sorts.

You will encounter the use of differentials combined with different mathematical rigour: they are defined precisely in differential geometry or non-standard analysis. Otherwise I personally consider them a helpful notation, for example when using separation of variables - the precise definition is much more unwieldy.

mvw
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  • So if I consider a volume element, in cartesian coordinates for example, The correct notation is $d^3V = dxdydz$ and not $dV = dxdydz$ as we can see everywhere ? – user1234161 Oct 10 '14 at 11:41
  • I have seen both notations used. The first one is more precise, but if you integrate in 3D all day, I can understand that authors drop the dimension. Another variant is $d^3\textbf{x}$. – mvw Oct 10 '14 at 11:48
  • Also, any order differentials have the same meaning of "a small element of something" right ? So I can choose the one that suits me to describe my problem using their respective properties ? – user1234161 Oct 10 '14 at 11:48
  • That really depends on the author. For Cauchy $dx$ was finite and constant, but $\Delta x$ was infinite small and variable. Today I see $\Delta x$ more for finite differences. If you explain what you mean by your notation you should be safe. – mvw Oct 10 '14 at 11:52