On the basis of the general calculation formulas and programming methods proposed in the first two articles, a comprehensive validation of one to three dimensions including displacement field, temperature field, seepage field, sound field, electrostatic field (magnetic field), and potential flow field is provided by solving the differential equations of motion in elasticity, conduction equation, and wave equation (including steady-state and transient analysis). The given examples show the unique features of the manifold method based on independent covers, including the meshes of any shape and connection, the simulation of precise geometric boundaries and the accurate application of essential boundary conditions, high-order series approximation, and the application of analytical series near crack tips. Finally, summarize the entire article and propose a new name of "series manifold element (series element)".