A Type of Singular Points for a Transformation of Three Variables: A Dissertation Submitted to the Faculty of the Ogden Graduate School of Science in ... Department of Mathematics (Classic Reprint) - Softcover

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Excerpt from A Type of Singular Points for a Transformation of Three Variables: A Dissertation Submitted to the Faculty of the Ogden Graduate School of Science in Candidacy for the Degree of Doctor of Philosophy; Department of Mathematics

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 81 in the Tye - space, and the initial assumptions in Section 1 are such that P and P1 are non-singular points of S and 81, 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 con nected region with interior points in the cryz-space, and the two xyz - regions so obtained adjoin 81 and lie on the same side of that surface. Moreover the single valued inverse functions of x, y, 2 defined by the transformation in either of the regions adjoining 81 are continuous, and they have continuous first derivatives except possibly at points of the surface 81.

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