On number of sheets of coverings defined by a system of equations in $n$-dimensional spaces

The coverings are mostly used in geometry and analysis, and sometimes they are not given explicitly. The problem on defining of covering in concrete situation is substantive. Coverings arose in theory of manifolds, especially in connection with the system of equations. One of powerful methods in this direction is a theorem on implicit functions.
In the paper we study these questions in a necessary general form. Such a consideration lead the problems to the basic notions which were studied by classics of mathematics in last two centuries. By him it was analyzed the main points of the theory on behavior of manifolds of less dimensions in manifolds of higher dimensions. Defining of the notion of a curve in the plane is bright example showing how we can establish suitable properties of objects we deal with to get the necessary freedom of actions, does not avoiding simplest generality. Introducing of quadrable curves makes possible to develop an acceptable notion of the integral in the domains on the plane. But this is insufficient for establishing for example, the theorem of Fubini on repeated integrals in that form as in Lebesgue's theory. Here we rest to constraints brought by intersection of manifold with boundary. The useful formulation of this theorem is possible to get only in Lebesgue theory of integration. This is one of multiplicity of questions connected with behavior of manifolds of less dimensions. We show how some notions of the theory must be modified to avoid such difficulties. We establish that the generalization of a notion of "improper" surface integral in some different from the ordinary meaning, makes possible solve the problem in general.
In the present work we lead by such method the question on estimating of the number of sheets of covering to some metric relations connected with surface integrals.