在前两篇提出的通用计算公式、通用程序设计方法的基础上，通过求解弹性力学运动微分方程、传导方程、波动方程（含稳态和瞬态分析），给出了一至三维包括位移场、温度场、渗流场、声场、静电场（磁场）和势流场等算例进行全面验证，涵盖了任意形状和任意连接的网格、精确几何边界的模拟及本质边界条件的准确施加、高阶级数逼近、裂纹尖端附近解析级数的应用等独立覆盖流形法的特色功能。最后对全文进行总结，并提出“级数流形元（级数元）”的新名称。

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)".