Extension of the unit normal vector field from a hypersurface

TL;DR

This paper provides an elementary proof for the existence and uniqueness of extending the unit normal vector field from a hypersurface to its neighborhood as a unit gradient field, which is important in various applications.

Contribution

It offers a simple, elementary proof of the existence and uniqueness of the normal vector field extension as a unit gradient field.

Findings

Proof of existence and uniqueness established

Extension is shown to be a unit gradient field

Elementary approach simplifies previous methods

Abstract

It is important in many applications to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of the paper is to provide an elementary proof of the existence and uniqueness of such an extension.