Wikipedia:Reference desk/Archives/Science/2023 June 15

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June 15

Density multiplied by a distance

If I have a density of points per unit volume (P) which varies according to the value of a function y=f(x) in meters, what would the product dP.dy give, a density per unit area or what else? Malypaet (talk) 21:52, 15 June 2023 (UTC)

I'm not able to work out what dP.dy means, in particuler what y=f(x) is supposed to mean. What exactly is the relation between P and f(x)? Perhaps a specific example might help. NadVolum (talk) 22:00, 15 June 2023 (UTC)

Total electron content might be a suitable practical example. catslash (talk) 22:30, 15 June 2023 (UTC)

What is meant is surely $\rho (x)\,\mathrm {d} x$, or better $\int \rho (x)\,\mathrm {d} x$. In astronomy, this is called column density; in general area density in units of number of points per area (according to OP's question) or mass per area if $\rho$ is a mass density. --Wrongfilter (talk) 06:00, 16 June 2023 (UTC)

Are you asking for the dimension of ${\text{d}}P{\text{d}}y$? What does it mean that $P$ "varies according to" the value of $y$? Does it mean $P=y=f(x),$ or $P=g(y)=g(f(x))$ for some unknown function $g$? And what is the role of $x$? Does it represent some physical quantity?  --Lambiam 09:20, 16 June 2023 (UTC)

Yes, for example if "P" is an energy density in joule/m3, a function of wavelength "y" in meter. When you read that "P" (dP ?) is the energy density slice between "y" and "y+dy", represented by the product "P.dy", what does this mean ? And we see here that this product give energy by area in m2, as in column density to area density. Why not represent the energy density slice by "dP(dy)" or some thing else more appropriate, to stay in energy density by volume as a function of "dy" ? Malypaet (talk) 13:36, 16 June 2023 (UTC)

There is no such thing as a "density slice". Density is defined at every point and is a function of position, $\rho (y)$. Writing $\mathrm {d} \rho$ only makes sense if you're talking about the derivative ${\frac {\mathrm {d} \rho }{\mathrm {d} y}}$, the gradient or rate of change of density with position. As far as I can tell this is not what you mean. The product $\rho \,\mathrm {d} y$ has dimensions of energy per area (e.g. J/m²), no way around this. --Wrongfilter (talk) 14:28, 16 June 2023 (UTC)

Sory I wanted to write "energy density of a spectral slice between "y" and "y+dy", but as energy density "P" vary as a function of y, I was considering "dP" as a kind of slice. Malypaet (talk) 15:07, 16 June 2023 (UTC)

"Spectral slice"? Is your "y" supposed to be a frequency $\nu$??? No, probably a typo for "spatial slice"?--Wrongfilter (talk) 15:48, 16 June 2023 (UTC)

You did not explain the meaning and significance of the variable $x.$ Also, what is the dimension of $P$ (or is it the Greek upper-case letter $\mathrm {P}$ or lower-case letter $\rho$?)? First you said a density of points per unit volume, but now you have switched to energy density. Which is it?  --Lambiam 18:12, 16 June 2023 (UTC)

The name of the letter does not matter, here the point or energy density is the same. With this exchange I found the solution to my problem which was to know how to pose it. "x" doesn't matter, "P" is a function of "y" expressed in meters, but in a fourth dimension. In this case it is necessary to use the integral calculus for "y" varying from 0 to infinity, as that one remains in a density of volume with "P" .

Thank you all Malypaet (talk) 08:01, 17 June 2023 (UTC)