Singularities of Mappings of Euclidean Spaces

• Hassler Whitney
Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

We shall describe here some results and methods pertaining to the following general problem (details will appear elsewhere). Suppose a mapping f 0of an open set R in n-spaceE n into m-space E m is given (we shall write f: R n E m ). How can we alter f 0slightly, obtaining a mapping f with nicer and simpler properties? By the Weierstrass approximation theorem (generalized), we may require that f be analytic in R; if f 0 was r-smooth (had continuous partial derivatives through the r th order), we may require the partial derivatives of f through the r th order to approximate those of f 0 (we then call f anr-approximation). Now take any regular point p of f, that is, a point p such that f is of maximum rank v = inf (n, m) at p. (Equivalently, using coordinate systems inE n and in E m , the Jacobian matrix of f at p is of rank v.) Then, by the implicit function theorem, we may choose coordinates so that f has the form
$${y^i} = {x^i}\left( {i = 1, \cdots ,v} \right), {y^i} = 0\left( {i > v, if m > n} \right).$$
(1.1)

Keywords

Singular Point Null Space Smooth Manifold Stable Mapping Regular Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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