A type of singular points for a transformation of three variables

Abstract

near points of a special type at which the jacobian of the transformation vanishes. Let a particular one of the singular points in question be denoted by P, and let S denote the surface through P in the uvw-space defined by setting the jacobian of the transformation equal to zero. The point P and the surface S are transformed by (1) into a point P1 and surface S, in the xyz-space, and the initial assumptions in Section 1 are such that P and P1 are non-singular points of S and Si, respectively. A neighborhood of the point P is divided by S into two parts. In Section 2 it is shown that under the hypotheses of Section 1 each of these parts is transformed in a one-to-one way into a connected region with interior points in the xyz-space, and the two xyz-regions so obtained adjoin Si and lie on the same side of that surface. Moreover the single valued inverse functions of x, y, z defined by the transformation in either of the regions adjoining Si are continuous, and they have continuous first der'ivatives except possibly at points of the surface Si. In the first two sections the transformation is studied in a neighborhood of a particular singular point P. But most of the results there found can be extended to apply to a neighborhood of any surface consisting of singular points of the type of P. This is done in Section 3. The generalization is analogous to an extension of the well-known theorems on implicit functions due to Bolza.t The methods of Sections 1-3 are applicable to n-dimensional spaces with but little more than the necessary changes in notation.