Utilizing a simple mathematical relation and Stoke’s theorem, the geometric-mean transmittance and total absorptance between an infinitesimal area and a finite planar element is reduced to a line integral around the planar element and an area integral. A concept of fundamental solutions is introduced. These are solutions in which the finite areas are horizontal right triangle with three specific orientations. Based on superposition, solutions for arbitrary finite areas are shown to be readily generated algebraically from these fundamental solutions. The geometric-mean transmittance and total absorptance between two finite areas are reduced to single numerical integrations, thus reducing much of the mathematical complexity. Fundamental solutions for mean beam lengths of the geometric-mean total absorptance in the weak-band, strong-band and very-strong-band limits are generated analytically in closed form. Based on the existing one-dimensional wide band correlation, these limiting expressions are shown to be sufficient for the calculation of the geometric-mean total absorptance at all optical thicknesses. A sample calculation is presented.

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