Editing Lambert's cosine law (section)

== Relating peak luminous intensity and luminous flux ==

In general, the [[luminous intensity]] of a point on a surface varies by direction; for a Lambertian surface, that distribution is defined by the cosine law, with peak luminous intensity in the normal direction. Thus when the Lambertian assumption holds, we can calculate the total [[luminous flux]], <math>F_\text{tot}</math>, from the peak [[luminous intensity]], <math>I_{\max}</math>, by integrating the cosine law:
<math display="block">\begin{align}
F_\text{tot} &= \int_0^{2\pi} \int_0^{\pi/2} \cos(\theta) \, I_{\max}\, \sin(\theta)\,d\theta\,d\phi\\
&= 2\pi\cdot I_{\max}\int_0^{\pi/2}\cos(\theta)\sin(\theta)\,d\theta \\
&= 2\pi\cdot I_{\max}\int_0^{\pi/2}\frac{\sin(2\theta)}{2}\,d\theta
\end{align}</math>
and so
:<math>F_\text{tot}=\pi\,\mathrm{sr}\cdot I_{\max}</math>
where <math>\sin(\theta)</math> is the determinant of the [[Jacobian matrix]] for the [[unit sphere]], and realizing that <math>I_{\max}</math> is luminous flux per [[steradian]].<ref>Incropera and DeWitt, ''Fundamentals of Heat and Mass Transfer'', 5th ed., p.710.</ref> Similarly, the peak intensity will be <math>1/(\pi\,\mathrm{sr})</math> of the total radiated luminous flux. For Lambertian surfaces, the same factor of <math>\pi\,\mathrm{sr}</math> relates [[luminance]] to [[luminous emittance]], [[radiant intensity]] to [[radiant flux]], and [[radiance]] to [[radiant emittance]].{{citation needed| date=June 2013}} Radians and steradians are, of course, dimensionless and so "rad" and "sr" are included only for clarity.

Example: A surface with a luminance of say 100&nbsp;cd/m<sup>2</sup> (= 100 nits, typical PC monitor) will, if it is a perfect Lambert emitter, have a luminous emittance of 100π lm/m<sup>2</sup>. If its area is 0.1 m<sup>2</sup> (~19" monitor) then the total light emitted, or luminous flux, would thus be 31.4 lm.