Based on a simple mathematical approach proposed in the paper, we demonstrate a rigorous computation of the Christoffel symbols and the Riemann tensor that obviously have a regular geometric dimension, which is extremely important in solving a huge class of purely physical problems. As examples, we consider four orthogonal coordinate systems, two of which are spherical and cylindrical, i.e. standard for describing any course of tensor analysis, and the other two are parabolic and orthogonal two-dimensional coordinate systems, for which the Christoffel symbols, the Laplace operator, and Riemann and Ricci, whose all components automatically have the correct geometric dimensions, are calculated. A number of physical applications of the described mathematical formalism are demonstrated. An example of a nonorthogonal two-dimensional coordinate system is considered, with the help of which a detailed calculation of the Christoffel symbols of both kinds is given, and an expression is found for the Laplace operator with application to the problems of elasticity theory and hydrodynamics.