2
$\begingroup$

I'm reading a text on ray tracing. There is this section about radiometric quantities where radiance is defined as

$L = \frac{d^2\Phi}{dA cos\Theta d\omega}$

$\Phi$ is the radiant flux

$\Theta$ is the solid angle (sr) subtended by the observation or measurement

$\omega$ is the incidence angle measured from the surface normal

This is just one of many equations using $d$ and $d^2$. I'm pretty sure that $d$ has something to do with differential equations. I already read some texts on differential equations but I still don't understand the meaning of $d$ and $d^2$ in this context.

Can someone explain this to me or point me to some reference/resource/book whatever?

Especially the $d^2$ puzzles me.

  • 2
    This $n$otation appears in the Wikipedia ( http://en.wikipedia.org/wiki/Radiance ) and with $m$ore detail i$n$ the German version (entry *Strahldichte* htt$p$://de.wikipedia.org/wiki/Strahldichte ) in the integral form: $$\Phi = \int_{\Omega} \int_A L_{\Omega}(\beta, \varphi) \cdot \co$s$(\beta) \math$r$m{d}A \cdot \mathrm{d}\Omega = \int_{\Delta\beta} \int_{\Delta\var$p$hi} \int_A L_{\Omega}(\beta, \varphi) \cdot \cos(\beta)\sin(\beta) \cdot \mathrm{d}A \, \mathrm{d}\beta \, \mathrm{d}\varphi$$ in accordance with the clarification by Raskolnikov.2011-01-13

1 Answers 1

3

This is fundamental notation in differential calculus. I suggested you pick up a book on the subject and read up; you won't regret it, as its very useful knowledge whatever you do!

In your specific cases, the expression is in fact a second-order partial derivative. The ds should be written in the curly style - this may be a fault of wherever you saw the expression from.

Here are some of the basics of notation, to get you started.

$\frac{dy}{dx}$ = the derivative (rate of change) of $y$ with respect to $x$. (1st order derivative)

$\frac{d^2y}{dx^2}$ = the derivative (rate of change) of $dy/dx$ with respect to $x$. (2nd order derivative, also called curvature)

Note: a common elementary mistake is to treat differentials as fractions. They are indirectly related, but do not treat them the same.

  • 0
    Thanks for the explanation. I think you're right. I really need to brush up my differential calculus skills to fully understand radiometry I guess. Can you recommend a book to me?2011-01-14