Equating Units in Plank's Blackbody Equation

[novop]

Homework Statement

My question is, why, when changing the variable from "v" to "lambda", does the $$d\lambda$$ need to be included? Why can't a simple substitution c = v * lambda do the trick?

 

[vela]

Because $\varepsilon_B$ is a density. If it's in terms of frequency, its units are something per unit frequency. Similarly, if it's in terms of wavelength, its units are something per unit wavelength. So you can't simply equate the functions. But the units of $\varepsilon_B(\lambda,T)\,d\lambda$ and $\varepsilon_B(\nu,T)\,d\nu$ are the same, so you can equate those.