Transfer Equation in General Curvilinear Coordinates

The differential operator of the monochromatic polarized radiative transfer equation is studied in case of statistically homogeneous turbid medium in Euclidean three-dimensional space, with arbitrary curvilinear coordinate system defined in it. An apparent rotation of the polarization plane along the light ray with respect to the chosen polarization reference plane generally takes place, due to purely geometric reasons. Using methods of tensor analysis, analytic expressions for the differential operator of the transfer equation depending on the components of the metric tensor and their derivatives are found. Considerable simplifications take place if the coordinate system is orthogonal. As an example, the differential operator of the vector radiative transfer equation in both elliptical conical coordinate system and oblate spheroidal coordinate system is written down. Nonstandard parameterization of standard elliptical conical coordinate system is proposed.