Abstract
Most of the literature on differential geometry presents the coefficients of the 1st and 2nd fundamental form to simplify the calculation of the curvatures on a surface and also to obtain other important information, such as the area of a surface. In this text, as the interest lies on the generalization of the idea of surface (hypersurface), such simplification by the use of these coefficients was not possible, in view of the complexity of the mathematical operations involved in the calculation of curvatures (if n = number of variables in the implicit function of the surface> 3), opting to use the linear operator –DNp. To facilitate the calculation of the normal vector the surface was described as the graph of a differentiable function f: Rn-1→R. The example of the sphere thoroughly illustrates the procedure of calculating the main curvatures of a surface in the space R3, and such procedure is extensive to the space Rn. Some examples from the engineering area in which the variables of the implicit functions of the hypersurfaces are random were solved, and the results of the main curvatures, calculated by the proposed procedure, were exact and coincident with those provided in the literature.