Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

(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


share|improve this question
add comment

1 Answer

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.

share|improve this answer
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.