Area–Solid Angle Integral: Area–Solid Angle Integral: A Tensor-Based Framework Unifying Engineering Flux Laws

The \emph{Area--Solid Angle Integral} (also referred to as the \emph{Alpha--Omega integral} or \emph{$\mathrm{A}\Omega$ integral}) is introduced as a tensor--based generalization of classical flux integrals. It provides a single compact framework that encompasses a broad family of laws --- including Gauss’s law, Fourier’s law, Darcy’s law, Ohm’s law, and the radiometric throughput factor --- as special cases. The kernel $\vec n^{\top}\mathbf T(\vec r,\vec\omega)\,\vec\omega$ combines the surface--direction factor $\vec n\cdot\vec\omega$ with the action of the physical transformation tensor $\mathbf T(\vec r,\vec\omega)$. In isotropic or diagonal cases this reduces to a simple product of geometry and physics, making explicit why flux laws across diverse domains share a common surface--direction structure. The formulation is intended both as a didactic bridge between classical flux integrals and pixel--level radiometry, and as a versatile modeling tool for anisotropic transport, multiphysics coupling, and wave phenomena.