Differentiable Germs and Catastrophes

This book gives a fairly elementary introduction to the local theory of differentiable mappings and is suitable as a text for courses to graduates and advanced undergraduates.

Theodor Bröcker (Author), L. Lander (Translated by)

9780521206815, Cambridge University Press

Paperback, published 17 July 1975

188 pages
22.9 x 15.2 x 1.1 cm, 0.29 kg

These notes give a fairly elementary introduction to the local theory of differentiable mappings. Sard's Theorem and the Preparation Theorem of Malgrange and Mather are the basic tools and these are proved first. There follows a number of illustrations including: the local part of Whitney's Theorem on mappings of the plane into the plane, quadratic differentials, the Instability Theorem of Thom, one of Mather's theorems on finite determinacy and a glimpse of the theory of Toujeron. The later part of the book develops Mather's theory of unfoldings of singularities. Its application to Catastrophe theory is explained and the Elementary Catastrophes are illustrated by many pictures. The book is suitable as a text for courses to graduates and advanced undergraduates but may also be of interest to mathematical biologists and economists.

1. Foreword
1. Germs with constant rank
2. Regular values
3. Construction of differentiable maps
4. Germs and jets
5. The division theorem
6. The preparation theorem
7. Symmetric germs
8. Mappings of the plane into the plane
9. Boardman-Thom singularities
10. The quadratic differential
11. Finitely determined germs
12. Some elementary algebraic geometry
13. Tougeron's theory
14. The universal unfolding of a singularity
15. The seven elementary catastrophes
16. Proof of the main theorem on universal unfoldings
17. Pictures of the seven elementary catastrophes
Further reading
Indices.

Subject Areas: Topology [PBP], Geometry [PBM]