Transmittance: Conflicting definitions?

[SMc21]

Not sure if this is in the right section, but I'm not sure where else it would fit.

I'm currently researching a variety of optics-based topics, and I'm a bit confused by what appear to be some conflicting definitions of transmittance. I've seen the following:

1) It's the ratio of monochromatic flux (i.e. flux per unit wavelength, in W nm^-1) transmitted through a medium to the monochromatic flux incident upon the surface. Mathematically:

T(λ) = $\frac{\Phi^{λ}}{\Phi_{0}^{λ}}$

2) It's the ratio of monochromatic irradiance (in W m^-2 nm^-1) transmitted to incident monochromatic irradiance. Mathematically:

T(λ) = $\frac{E^{λ}}{E_{0}^{λ}}$
Now, if flux is constant across the surface of the medium, then obviously E$^{λ}$ = $\frac{\Phi^{λ}}{S}$, where S is the surface area of the medium receiving the light. Then Equation 2 reduces to Equation 1, and the definitions are equivalent.

However, in the general case (not assuming that flux is constant over the surface area), E$^{λ}$ = $\frac{d\Phi^{λ}}{dS}$. Then Equation 2 becomes

T(λ) = ($\frac{d\Phi^{λ}}{dS}$) $/$ ($\frac{d\Phi_{0}^{λ}}{dS}$)Thus, in the general case, transmittance is either the ratio of transmitted monochromatic flux to incident monochromatic flux, or the ratio of their derivatives with respect to surface area.
Are these two definitions not at odds with each other? Do we assume that the two ratios described above are equivalent? If so, what justifies that assumption? Or am I missing something in my understanding here?

Any help is greatly appreciated!

 

[Simon Bridge]

It may well be that there are situations where the definitions are at odds with each other - it is not terribly uncommon for a definition to change with the context: there are only 1,019,729.6 words in the English language and the easy ones are already taken.

You should go back to the context of the different definitions and see if the author has derived an equivalent maths that is valid for the specific circumstances they are talking about.