Abstract: The locally existed orthogonal parameter coordinates of smooth surface in differential geometry has been related to nonholonomic basis in tensor analysis. This provides a new method for tensor field theory on surfaces and their neighborhoods termed as the nonholonomic basis theory based on the orthogonal system of the principal directions of the surface. This field theory method is based on the parameter coordinates of the base surface with respect to the principle directions, and then performs spatial extension in the normal direction of the base surface to obtain a complete orthogonal system in the base surface neighborhood. Introducing the nonholonomic basis theory, it can be obtained that all non-zero Christoffel symbols directly correspond to the principal curvatures or geodesic curvatures of the base surface or the local surface. Consequently, the resulting components of all kinds of differential operators of any tensor field with respect to the principle directions and the normal direction contains only the physical quantities and the curvatures of the surface. This not only clearly shows the relationship between the geometric characteristics of the surface and the physical quantity/physical process, but also the component expression obtained is the simplest in form. On the other hand, classical orthogonal systems such as cylindrical coordinate system and spherical coordinate system also belong to the orthogonal system based on the principal directions of the surface, so the method in this paper can unify the tensor field analysis of the related orthogonal systems. As an application, the expressions of vorticity, normal gradient of vorticity and deformation rate tensor on deformable surfaces, component equations of fluid boundary layer equations on surfaces, and related conservation law equations for curved media are derived.