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I have just started basic quantum mechanics and have come across this expression for energy density of blackbody radiations; Planck's formulation. $$ u_{\nu} \ d\nu = g(\nu) \ \langle E \rangle \ d\nu $$

The textbook says that:

To model the absorption and emission of electromagnetic radiation by these oscillators, he argued that the spectral energy density of the black body radiation over a range of frequency dv is nothing but the product of the number of oscillators having a frequency between v and v + dv , g(v) and the average energy ⟨E⟩, of an oscillator having a frequency v :

What I don't understand is: if u(v) is the energy density (per unit v), isn't $u(\nu)\ d\nu$ the total energy instead of energy density? And one more question; if $g(\nu)$ is the number of oscillators having frequency in the range $d\nu$, and $⟨E⟩$ is the average energy; then isn't their product the total energy instead of being the energy density, and if that is supposed to be so then why do we have a $d\nu$ part on the right side of equation also?

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$g(\nu)$ would be the number density of oscillators per unit frequency at a frequency $\nu$.

$u(\nu)\ d\nu$ would be the energy density in a frequency band $d\nu$ at a frequency $\nu$.

The $d\nu$ is written on both sides of the equation since otherwise you are saying $0 = 0$, because no oscillators have a frequency of exactly $\nu$.

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