Here are the chapters of the book:. This treatise is not a run-of-the- mill mathematics book. It offers a fresh point of view on some of the basic infrastructure of mathematics.
It unifies all aspects of finite-dimensional spaces by stressing the interplay between algebra, geometry, topology, and analysis. Most of the proofs are original and many concepts and results cannot be found elsewhere. The approach is uncompromisingly coordinate-free and R n - free when dealing with concepts.
However, there is a chapter on how to handle general coordinate systems efficiently when dealing with special situations.
This treatise cannot be understood by "sampling" miscellaneous sections or chapters. The reader must first become familiar with the basic terminology and notation presented in the Introduction and the preliminary Chapter 0. This treatise should be a suitable textbook for core courses at the advanced undergraduate and beginning graduate level. Also, it should be of use to theoretically inclined scientists and engineers who wish to use contemporary mathematics as a conceptual tool. The first volume of this treatise was first published in The present version, posted on my website in , is a corrected reprinting.
The second volume is not yet completed, but preliminary versions of various chapters are posted and frequently updated on this website. Therefore, it is the result of a task similar to that undertaken by Bourbaki, albeit on a limited scale. I have also been accused of failing to use "standard" notation and terminology. I plead guilty again. There is no "standard" terminology for many of the concepts I describe.
For others, the "standard" terminology is all too often misleading, illogical, obscure, ungrammatical, clumsy, archaic, or downright stupid. In these cases, I have not hesitated to introduce my own terminology. However, in the Introduction and the Notes to each section I mention and comment on other terminology. I consider this treatise to be the most important work of my life. I , pages I Ind. The plan is to publish this volume in its entirety in the future. Here are preliminary manuscripts of the Chapters 0, 1, 2, 3, and 4.
Preliminaries, 8 pages.
Differential Calculus I I , 17 pages. Manifolds in a Euclidean Space , 33 pages.
Volume Integrals , 47 pages. A Chain Rule of Order n should state, roughly, that the composite of two functions that are n times differentiable is again n times differentiable, and it should give a formula for the n'th derivative of this composite. Many textbooks contain a Chain Rule of order 2 and perhaps 3, but I do not know of a single one that contains a Chain Rule of arbitrary order n with an explicit and useful formula.
The main purpose of this pape r i s to derive such a formula. In this paper, it is shown how the concept of a Boolean algebra can be based on that of a monoid. Each entry will be preceded by an abstract. The entries will be updated periodically. Table of Contents A. Pure Mathematics. This essay is intended not only to help professors better understand their own role, but also to help the public at large better appreciate this role.
In this case we have! Orientations In order to explain orientations on manifolds we first start with orientations of finite-dimensional vector spaces. M be an injective immersion. The text is unusual in structure and emphasis. We can now apply Theorem 6. It will be for some of you the first course in which you are expected, not to calculate answers, but to give proofs. In this setting Whitney established some improvements of Theorem 3.
Although the essay is written from the point of view of a professor of mathematics, its essence should apply to professors in any field. By conceptual infrastructure of mathematics I mean the concepts, the terminology, the symbols, and the notations that mathematicians and people who apply mathematics use in their daily professional activities.
This infrastructure was and still is being developed in four stages. Here I make an attempt to describe these stages. It was first published in and has become the standard reference work in the field. It was reprinted in , translated into Chinese in , and again reprinted in This is an account of the efforts that went into the creation of this work.
This paper was published in the Journal of Elasticity 70, A part of it is included in the beginning of the edition. The term "Natural Philosophy" was used from the 17th to the middle of the 19th century for what is now called "Natural Science". About 50 years ago, the term was revived by Clifford Truesdell. I will describe why he revived this term.
Here is his description of the meaning of the term: "In modern natural philosophy, the physical concepts themselves are made mathematical at the outset, and mathematics is used to formulate theories". Truesdell provided the leadership that led to the foundation of the Society for Natural Philosophy" in I will describe its purpose and my hopes for the future of natural philosophy.
I list 8 aspects of mathematics and give my take on their relative importance. Two of these are the creation of new concepts by abstraction and the clarification of old ones. The former is illustrated by the creation of the concept of a monoid and the latter by the clarification of the concept of volume. This is the text of a lecture before the mathematics teachers of the Catholic Schools in the Diocese of Pittsburgh.
First, I make the distinction between arithmetic and true mathematics, starting with geometry and algebra. For the latter, rote memorization is deadly while conceptual understanding and problem solving ability are essential.
I illustrate this insight by a section entitled The art of avoiding unnecessary calculation and a section on The Theorem of Pythagoras. This is a description of my life from my birth in and my arrival in Pittsburgh in The emphasis is on my education in Germany, France, and the United States and on the development of my interest in mathematics.
This is the text the lecture given by Clifford Truesdell in April at the meeting of the Society for Natural Philisophy on the occasion of my retirement from teaching. The emphasis is on the situation in the Graduate Institute of Applied Mathematics at Indiana University both before and after my arrival there in This is a proposal for a system, based on the internet and websites, that is faster, cheaper, and more efficient than the traditional one.
Table of Contents. Framework for Rheology.
If you are interested in taking this course but can't make the time slot, please let me know; I will try to change the time to accommodate as many students as possible, subject to my own schedule and room availability. Prerequisites A background in the basic topology of smooth manifolds e.
Ma c , as well as an understanding of the fundamental group from algebraic topology e. Ma a or Ma a. More advanced knowledge of algebraic topology e. Ma bc is not needed. Grading Grades will be based on weekly homework assignments. In particular, there will be no exams in this course. The weekly homework will be due in class on Fridays, at the beginning of class.
Here "beginning of class" will be interpreted generously, say up to 10 minutes after I start, so no need to rush if your running a bit late. Homework turned in after that, up through the beginning of class on Monday, will be graded but only count for half-credit. Beyond this, late homework will not be accepted, except in certain extreme circumstances, in which case you must ask me for an extension prior to the due date. However, your lowest homework grade will be dropped. If you won't be in class on Friday, turn your homework into my box outside the math department office at least 15 minutes prior to the start of class; late-but-not-too-late HW can be put in the same place.
Texts I will not be following any particular text closely, and there is no required text for this course.