Categories in Continuum Physics Lectures given at a Workshop held at
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1174 Categories in Continuum Physics Lectures given at a Workshop held at SUNY, Buffalo 1982
Edited by F. W. Lawvere and S. H. Schanuel
Springer-Verlag Berlin Heidelberg New York Tokyo
Editors
F. William Lawvere Stephen H. Schanuel State University of New York at Buffalo Mathematics Department, 106 Diefendorf Hall Buffalo, N.Y. 14214, USA
Mathematics Subject Classification (1980): 18-xx, 53-xx, 58-xx, 70-xx, 76-xx, 80-xx ISBN 3-540-16096-5 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16096-5 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging-in-Publication Data, Categories in continuum physics. (Lecture notes in mathematics; 1174) Bibliography: p, 1. Field theory (Physics) - Addresses, essays, lectures. 2, Continuum mechanics-Addresses, essays, lectures, I. Lawvere, F, W.II. Schanuel, S. H. (Stephen Hoel), 1933-, III. Series: Lecture notes in mathematics (Springer-Verlag); 1174.
0A3.L28 no. 1174 [OC174.46] 510 s [530] 86-1806
ISBN 0-387-16096-5 (U.S.)
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Pre f ace
The success of the May 1982 Workshop which resulted in this volume was due in large measure to the active and friendly participation of all who attended, to the valued assistance of Rosemary Marciniak and Judy Bittner, and to the financial assistance granted by
SUNY
Buffalo.
We thank all these, and
especially thank Fatima Lawvere, whose dedicated efforts brought about the atmosphere which all enjoyed. For the preparation of this volume, the editors give thanks to Gail Berti for her excellent typing, and to the National Science Foundation for financial support. F. William Lawvere Stephen H. Schanuel
CON
TEN
T
S
F. W. Lawvere
Introduction
W. Noll
Continuum Mechanics and Geometric
Integration Theory . . . .
17
Williams
Structure of Continuum Physics
30
Chen
On Differentiable Spaces
38
Cartesian Closed Categories and
Analysis of Smooth Maps .
43
Introduction to Synthetic Differential
Geometry, and a Synthetic Theory of
Dislocations . . . . . .
52
Synthetic Reasoning and Variable Sets
69
Recent Research on the Foundations
of Thermodynamics . . . . . . • .
83
W.
o.
K-T. A. A.
G.
FrtHicher Kock
Reyes
B. D. Coleman and D. R. Owen D. R. Owen M. Feinberg and R. Lavine S. H. Schanuel
1
Global and Local Versions of the
Second Law of Thermodynamics . . .
100
Thermodynamics and the Hahn-Banach
Theorem . . . .
115
What is the Length of a Potato?
An Introduction to Geometric
Measure Theory .
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