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(It's mostly a calculus question and has little to do with optics.)

I'm reading a book on Computer Graphics, Realistic Image Synthesis Using Photon Mapping by Henrik Wann Jensen, and I can't completely understand how to derive the radiance equation given in Chapter 2.

According to the book, the spectral radiant energy $Q_{\lambda}$, in $n_{\lambda}$ photons with wavelength $\lambda$ is

$$Q_{\lambda}=n_{\lambda}\frac{h \; c}{\lambda}\;,$$

where h is Planck's constant. Hence


Radiant flux $\Phi$ is the time rate of flow of radiant energy:


And radiant flux area density is


The radiant intensity $I$ is the radiant flux per unit solid angle $d\omega$:


So the radiance $L$ is the radiant flux per unit solid angle per unit projected area:


My question is: Could anyone explain how to derive the last integral, and why it is not


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up vote 0 down vote accepted

This is an error in the book. By the first three equations, the units of

$$ \frac{n_\lambda}{\mathrm dt}\frac{hc}{\lambda}\mathrm d\lambda $$

are those of $\Phi$, and $\cos\theta\,\mathrm dA\,\mathrm d\omega$ is the same on both sides, so there's an unmatched inverse unit of length from $\mathrm d\lambda$ in the denominator on the right-hand side.

Your version of the expression looks OK to me.

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Thanks. Actually I've found other confusing formulas after posting that, which I'm quite sure are mistakes. So is this one, probably. – Stupident Nov 2 '12 at 15:06

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