To the Question of Gauss’s Curvature in n-Dimensional Euclidian Space

An alternative way of calculating the Gauss curve of the surface in the Cartesian coordinates in a three-dimensional case has been proposed, and its generalization in the ndimension case of measuring Euclidean space has been given.


Introduction
The issue that is touched upon in this work is directly related to differential geometry, and it is related to the common properties of smooth surfaces. It must be said that in this direction there are a lot of original publications and monographs (see, for example, Mac Connell (1957), Rashevsky (1967), Fihtenholtz (1969), Deepmala (2015), Kisi, et all (2016), Sezgin Buyukkutuk, et. all (2016, Laurian-Ioan, et. all (2017), Vandana, et. all (2016), Laurian-Ioan, et. all (2017)) in which all the basic properties of surfaces are presented in great detail and with strict mathematical evidence, including detailed calculations of their curvature radius R .
However, the approach that will now be demonstrated is very different from the methodical approach outlined in the classical monograph Rashevsky (1967), and it is based on a qualitatively different assumption.
The essence of it is this. As is customary in differential geometry, all calculations are reduced to an analysis of dependency not by the curvature radius R , but by the curvature k associated with R a simple inversion 1 k R  conversion, in the case of a flat task. In a three-dimensional case, the curvature is usually introduced in a slightly different way, namely, in the form of In this work, we will approach this problem, based on the idea of a large-scale theory, which allows using a simple algorithm to offer a very compact and strict conclusion for the curvature of the surface K , allowing to bring its generalization to case n  of measuring Euclidean space. This task will be part of the first part of this work. In the second part of the article, we will give a simple summary of the proposed approach for the case of arbitrary curved coordinates.

Euclidean n Is a Dimensional Space
Recall at first some commonly used formulas from the theory of differential geometry. As the main calculation formula for solving various tasks from mechanics and physics related to the theory of two-dimensional curved motion, as a rule, the expression is used where   y y x  is the trajectory of the body in the coordinates xy  and strokes traditionally indicates the appropriate derivatives.
The formula (1) is easily derived from a simple large-scale identity  is the element of the body angle, and dS  is the surface element corresponding to this element of the body angle, with the radius of curvature R .
This means that the curvature in three-dimensional space we can enter as a fraction It is quite clear that by following the algorithm of composing expressions (2) and (3) they can be easily summarized in case of space of arbitrary dimension n , and write down the general expression for curvature in the form of where 1 n d   is body angle element covering the surface element An element 1 n d  calculates perfectly trivial, as well as the hypersurface element 1 n d   , which we will now demonstrate on a simple example of three-dimensional Euclidean space.
With this goal let's use a simple geometric interpretation. Indeed, let the surface be set by a clear equation Then from the decomposition the radius -vector on a single orthogonal basis ,, i j k in the accordance to the where the cuts are introduced , This means that the desired curvature of the surface according to (3) taking into account (6) and (13) Note that the formula (14) has exactly the same look as Rashevsky (1967). That is the approach we have outlined is quite correct.
This is reflected in the formula (15). The threedimensional hypersurface element according to (6) in the accordance with the designations (4) is Hence due to the determination (4) curvature will be determined privately from expressions (15) and (18) Thus, the curvature computation algorithm is generally absolutely clear, and we can summarize the formulas (13), (15) and (19) and set the surface with a clear equation that instead of (14) and (19) Formula (23) adequately responds to the question of the curvature radius of any n  dimensional Euclidean surface set by an explicit equation (21) set at the beginning of the article. As an example, consider the three-dimensional of the sphere of radius b , the implicit equation of which will be set in the form of 2 2 2 2 x y z b    .
Therefore, for angular variables, we have