This is a basic technique used frequently in going from the general coordinate-free radiative transfer equation (RTE) to the RTE formulated for the plane-parallel atmosphere geometry (see Liou, Introduction to Atmospheric Radiation, 2002: Sec. 1.4.4), however I don't follow how it's done and the details are not provided (assumed 'obvious')
the general coordinate-free RTE can be written as a simple 1st order DE:
$$ \frac{dI(s)}{\gamma(s)ds} = -I(s) + J(s) \qquad (1) $$
where the path is $s$, and a differential change along path = $ds$. I suppose it doesn't even really matter if you know what the rest of the individual terms mean, if you're just thinking in terms of solving a DE.
Now we want to go to a specific geometry, with vertical zenith = z, zenith angle = $\theta$, and azimuth angle = $\phi$.
So now restricting paths to be (radial) straight lines in the direction $s(\theta,\phi)$ I believe we can write
$ z(\theta) = s(\theta,\phi)\cos\theta $
so that $ dz = ds\cos \theta \\ ds = dz/\cos\theta$
So now I substitute for $ds$ in (1) using above to get $$ \cos\theta \frac{dI(s)}{\gamma(s)dz} = -I(s) + J(s) \qquad (2) $$ Now we've gotten to my first point of confusion. Here they will often simply re-write Eq. (2) (see Liou Eq. 1.4.10) as, $$ \cos\theta \frac{dI(z;\theta,\phi)}{\gamma(z;\theta,\phi)dz} = -I(z;\theta,\phi) + J(z;\theta,\phi) \qquad (3) $$
Is that legal, to simply replace the $s$ variable in all the function arguments with $z;\theta,\phi$ ??
Isn't it true that $ s(z(\theta)) $ (noting $z(\theta)$ is independent of $\phi$ in this geometry)? Then don't we need to substitute $s = z/\cos\theta $??
Now, the next step taken (and my second point of confusion) is they define
$ \mu = \cos \theta$
and then just re-write Eq. (3) as
$$ \mu \frac{dI(z;\mu,\phi)}{\gamma(z;\mu,\phi)dz} = -I(z;\mu,\phi) + J(z;\mu,\phi) \qquad (4) $$ So again, can you just make that substitution in the function arguments $\theta \to \mu$? What about $ \theta = \arccos \mu $??
Confusion Point #3: From Eq. (4) we now introduce another new variable (the optical depth $\tau$) as $$ \tau(z) = \int_z^{\infty} \gamma(z';\mu,\phi)dz' $$ which is measured downward with $\tau(\infty) = 0 $ so we have $ d\tau(z) = -\gamma(z;\mu,\phi)dz $
(at least I think that's where the negative sign comes from...?) and for the final hurrah, they now write [Liou, Eq. 1.4.22] $$ \mu \frac{dI(\tau;\mu,\phi)}{d\tau} = -I(\tau;\mu,\phi) + J(\tau;\mu,\phi) \qquad (5) $$ Based on my above confusion regarding the previous changes-of-vars made inside the function arguments, you get bet I'm baffled on how we 'clearly' (grrrrrr) go from $ z \to \tau $ inside the arguments!!
I guess all 3 of my 'confusion points' are related to how we generally do these change-of-variables inside functions, when we may not know the specific form of the functions involved (for instance $I(...)$ and $J(...)$ in above Eqns). Can someone please help me understand how this is done correctly?