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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.

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$d$ is a notation for differential. In your formula, there is a $d^2$ because in the numerator there are two differentials. The formula is thus the second derivative of a function with respect to two variables. – Raskolnikov Jan 13 '11 at 20:26
This notation appears in the Wikipedia ( ) and with more detail in the German version (entry Strahldichte ) in the integral form: $$\Phi = \int_{\Omega} \int_A L_{\Omega}(\beta, \varphi) \cdot \cos(\beta) \mathrm{d}A \cdot \mathrm{d}\Omega = \int_{\Delta\beta} \int_{\Delta\varphi} \int_A L_{\Omega}(\beta, \varphi) \cdot \cos(\beta)\sin(\beta) \cdot \mathrm{d}A \, \mathrm{d}\beta \, \mathrm{d}\varphi$$ in accordance with the clarification by Raskolnikov. – Américo Tavares Jan 13 '11 at 20:51
up vote 3 down vote accepted

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.

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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? – Jan Deinhard Jan 14 '11 at 8:19

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