Hassler Whitney Collected Papers pp 436-452 | Cite as

# Singularities of Mappings of Euclidean Spaces

Chapter

## Abstract

We shall describe here some results and methods pertaining to the following general problem (details will appear elsewhere). Suppose a mapping

*f*_{0}of an open set*R*in*n*-space*E*^{ n }into*m*-space*E*^{ m }is given (we shall write*f*:*R*^{ n }→*E*^{ m }*)*. How can we alter*f*_{0}slightly, 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*an*r-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 in*E*^{ 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
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## Bibliography

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## Copyright information

© Birkhäuser Boston 1992