= First angular moment of a radiance field
{title2=$\int\boldsymbol\omega L(\boldsymbol\omega)\,d\Omega$}
The \b[first angular moment of a radiance field] is the vector
$$
\mathbf M=\int_{S^2}\boldsymbol\omega L(\boldsymbol\omega)\,d\Omega,
$$
with units $\mathrm{W\,m^{-2}}$ for frequency-integrated <radiance>. It records a directional imbalance. Here $\boldsymbol\omega$ is the viewing direction toward the arriving light, as in <dipole anisotropy of a radiance field>; using propagation directions instead reverses the vector. With that convention change, this is the bolometric vector form of the <radiative flux>, an existing <radiation-field moment>.
Opposite ideal planar receivers measure this moment without assuming a pure dipole. Since $\max(0,x)-\max(0,-x)=x$, the cosine-response <irradiance> formula gives
$$
E(\mathbf n)-E(-\mathbf n)=\mathbf n\cdot\mathbf M.
$$
Three orthogonal receiver pairs therefore determine the three components. For original differences $(0.60,-0.20,0.30)\,\mathrm{W\,m^{-2}}$, the moment has those components and magnitude $0.70\,\mathrm{W\,m^{-2}}$. The derivation requires the same spatial field, stable gains and known angular response. The primary optical guide describes cosine-receptor characterization rather than assuming it perfect.<Integrating Sphere Theory and Applications by Labsphere>[§3.1.3, Cosine Receptor]{ref}
A zero first moment does not establish an <isotropic radiance field>. For example, two equal oppositely directed collimated beams have cancelling vector moments while remaining strongly directional. Conversely, any nonzero first moment rules out isotropy. The scalar all-direction integral $J=\int L\,d\Omega$ bounds the moment by $|\mathbf M|\leq J$ for nonnegative <radiance>, by the <triangle inequality> and $|\boldsymbol\omega|=1$.
Within the restricted <dipole anisotropy of a radiance field> model, $\mathbf M=(4\pi/3)L_0\mathbf b$ determines its dipole vector. More general angular structure is not reconstructed by these three numbers. Opposed receivers reveal the first moment, not the entire directional distribution.
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