The dimensions of electromagnetic fields on linear, planar, and volumetric observation manifolds are studied by rigorously deriving and counting all the significant modes to linearly represent the fields with great accuracy. The fields are received with either full scattering or azimuthal scattering only. Both the uni-polarized scalar wave case and the fully-polarized vector wave case are investigated. The ratio of the field dimensions between the two cases gives rise to what is usually called the polarization gain in the literature. By tackling the problem from a purely electromagnetics perspective, this work offers an alternative approach of assessing the MIMO degrees-of-freedom limit in addition to a heuristic approach utilized by a recent study. Involving almost no approximation during the analysis, the results here bring improvement in terms of rigor and accuracy over those from the heuristic approach. Additionally, by using some numerical examples, this paper also explains how the field dimensions and the polarization gains should be properly interpreted in the context of realistic MIMO applications.