Transcription of TOPOLOGY FROM DIFFERENTIABLE VIEWPOINT
{{id}} {{{paragraph}}}
Example: marketing
TOPOLOGY FROM DIFFERENTIABLE VIEWPOINT
TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT By John W. Milnor Princeton University Based on notes by David W. Weaver The University Press of Virginia Charlottesville PREFACE THESE lectures were delivered at the University of Virginia in December 1963 under the sponsorship of the Page-Barbour Lecture Foundation. They present some topics from the beginnings of TOPOLOGY , centering about L. E. J. Brouwer s definition, in 1912, of the degree of a mapping. The methods used, however, are those of differential TOPOLOGY , rather than the combinatorial methods of Brouwer. The concept of regular value and the theorem of Sard and Brown, which asserts that every smooth mapping has regular values, play a central role. To simplify the presentation, all manifolds are taken to be infinitely DIFFERENTIABLE and to be explicitly embedded in euclidean space. A small amount of point-set TOPOLOGY and of real variable theory is taken for granted.
Translating both hyperplanes to the origin, one obtains dfz. Before giving the actual definition, we must study the special case of mappings between open sets. For any open set U C Rk the tangent space TU, is defined to be the entire vector space Rk. For any smooth map f: U + V the derivatiue dfz ; Rk + R' is defined by the formula
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: