This website was created to publish some recent work of Walter Noll and his collaborators as PDF files. Only a part of this work has also been published elsewhere.

Each entry will be preceded by an abstract.

Any comments, reviews,
critiques, or
objections are
invited and should be sent to me by e-mail: wn0g@andrew.cmu.edu. The
entries will be updated
periodically.

A
short curriculum vitae of Walter Noll and a complete list of his
publication is available on the following link

**A.
General**

1) The Role of the Professor

2) The Conceptual Infrastructure of Mathematics

3) The Genesis of the Non-Linear Field Theories of Mechanics

4) On the Past and Future of Natural Philosophy

5) What is Mathematics all about

6) Mathematics should not be boring

7) Autobiographical Notes

8) The Naturalization and Apotheosis of Walter Noll

9) The Future of Scientific Publication

B. Mathematical Science

1) Five Contributions to Natural Philosophy

2) Plugs in Viscometric Flows of Simple Semi-Liquids

3) On the Theory of Surface Interactions

4) A Frame-Free Formulation of Elasticity

5) Basic Concepts of Thermomechanics

6) On the Concept of Force

7) Mathematical Structures of Special Relativity

8) Thoughts on Stress

9)Thoughts on Thermomechanics

10) Frame-Free Thermomechanics, by Brian Seguin

**C. Pure Mathematics**

1) Finite-Dimensional Spaces, Algebra, Geometry, and Analysis Vol. I

2) Finite-Dimensional Spaces, Algebra, Geometry, and Analysis, Vol. II

3) The Chain Rule for Higher Derivatives

4) Monoids, Boolean Algebras, Materially Ordered Sets,

5) Monoids

6) Isocategories and Tensor Functors

7) Geometry of Differentiable Manifolds

8) Linearly Induced Mappings between Cones of Quadratic Forms

9) On the Conjugacy of Orthogonal Groups

10) Scalar Algebras and Quaternions

**A.
General**

**1)** **The
Role of the
Professor **(1997), 3 pages.

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. Although the essay is written from the point of view of a professor of mathematics, its essence should apply to professors in any field.

2)**The Conceptual Infrastructure of
Mathematics **(1995), 5 pages.

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.

**
3) The genesis of the Non-Linear
Field Theories of Mechanics** (2002),

9 pages.

I
was co-author,
with Clifford Truesdell, *of The Non-Linear Field Theories of
Mechanics*. It was first published in 1965 and has become the
standard reference work in the field. It was reprinted in 1992,
translated into Chinese in 2000, and again reprinted in 2004. 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,
**23-30 (2003). A part of it is included in the beginning of the 2004
edition.

**
4) On the Past and Future
of Natural Philosophy **(2005), 13 pages.

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 1963.

Recently,
I wrote a
booklet entitled
"Five
Contributions to Natural Philosophy", dedicated to the memory of
Clifford Truesdell. I will describe its purpose and my hopes for the
future of natural philosophy.

PFNP

**
5) What is Mathematics all about **(2006), 9
pages.

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.

**6)
Mathematics
should not be boring
**(2003),
18 pages.

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.*

**
7)Autobiographical Notes **(1988), 17 pages.

This is a description of my life from my birth in 1925 and my arrival in Pittsburgh in 1956. The emphasis is on my education in Germany, France, and the United States and on the development of my interest in mathematics.

**
8) The Naturalization and Apotheosis of Walter
Noll, **by Clifford Truesdell, 1993, 18 pages.

This is the text the lecture given by Clifford Truesdell in April 1993 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 1953.

9) The Future of Scientific Publication (2008), 6 pages.

This is a proposal for a system, based on the internet and websites,
that is faster, cheaper, and more efficient than the traditional one.

B. Mathematical Science

**1)
Five
Contributions to Natural
Philosophy **(2004), 73 pages.

I
was co-author,
with Clifford Truesdell, *of The Non-Linear Field Theories of
Mechanics*. It was first published in 1965 and has become the
standard reference work in the field. It was reprinted in 1992,
translated into Chinese in 2000, and again reprinted in 2004.

The *Five
Contributions to Natural
Philosophy*
provide a blueprint for updating *The Non-Linear Field Theories of
Mechanics* and hence a guideline for future developments in the
field. It should be available next to *The Non-Linear Field Theories
of Mechanics* in every scientific library.

**Table
of Contents**.

[0] Introduction

[1] On the Illusion of Physical Space

[2] On Material Frame Indifference

[3]
Updating *The
Non-Linear Field
Theories of
Mechanics*

[4] The Theory of Simple Semi-Liquids, a Conceptual

Framework for Rheology

[5] Nematic Semi-Liquids

**2)
Plugs in
Viscometric Flows of
Simple
Semi-Liquids,** by Walter Noll and Brian Seguin (2007), 14 pages.

**Here
we apply
the theory of simple semi-liquids described in part [4] of the booklet 1) above in order to analyze all
the viscometric flows treated in the book Viscometric Flows of Non-Newtonian Fluids by
B.D.Coleman, H. Markovitz, and W. Noll, Springer-Verlag, 1966. Only
simple liquids are treated in this book. When dealing with simple
semi-liquids, plugs can occur. This paper gives a detailed analysis
of such plugs.**

**3) On
the Theory of
Surface
Interactions. **(2005),
14 pages.

I
use the term *interaction
*as an abstraction that applies, for example, to force-interactions,
torque-interactions, and heat-transfers.

If an
interaction is a *surface-interaction
*it
can be characterized, under suitable conditions, by a *contactor.*
In the case of force-interactions, torque-interactions, and
heat-transfers, the contactor is the field of *stress-tensors,
couplestress-tensors, *and* heatflux-vectors,* respectively.

The first to prove, making suitable assumptions, the existence of such a contactor was Cauchy in 1823, and there has been a large literature on this subject since then.

Here I
describe what I
consider to be
the present
situation.

**
4) A Frame-Free Formulation of Elasticity **(2005),
16 pages

As I pointed out
at the end of Sect.4 in [N2] of the booklet *Five Contributions to
Natural Philosophy***, **it should be possible to make the
principle of material frame-indifference vacuously satisfied by using
an intrinsic mathematical frame-work that does not use an external
frame-space at all when describing the internal interactions of a
physical system. In Sects.2 and 3 below I will do just that for the
classical theory of elasticity by specializing the treatment given in.
In Sect.4 I will do the same for hyperelasticity, i.e., elasticity
based on a strain-energy function. In Sect.5 I will comment on possible
restrictions on the
intrinsic response functions defined in Sects.2 and 4.

**
5) Basic Concepts of Thermomechanics** (2009), 21 pages

This paper is intended to serve as a model for the first few chapters of future textboods on continuum mechanics and continuum thermomechanics.

6) On the Concept of Force, (2007),
4 pages.

This
essay is a reaction to an article by Frank Wilczek entitled "Whence the
force F = ma ?" in the October 2004 issue of
Physics today and to the letters
referring to this article in the August 2005 issue.

**7)
Mathematical Structures of Special Relativity
by Vincent J.Matsko and
Walter Noll (1993), 239 pages.**

**
**This
textbook is
based on a course given several times by Walter Noll from about 1970 to
1993. The approach presented is intended to convey a deeper
understanding of the subject and an intuition for relativistic
phenomena. Such an understanding can be gained only by the modern
mathematics of sets, mappings, relations, and mathematical structures.
Here are the chapters of the book:

Table of Contents:

Introduction

**1.
Eventworlds**INTRC1

2. Timed Eventworlds

3. Flat Eventworlds

4. Classical Spacetimes

5. Minkowskian Spacetimes

6. World Momentum

7. Electromagnetism

Appendices

This is the content of a lecture given at the SIAM meeting in Philadelphia in May 2008.

TOS

9)
Thoughts on Thermomechanics (2008).

TOT

10) Frame-Free Thermomechanics (2010).

This is Brian Seguins doctoral thesis, submitted in May 2010

BT

(11) Elastostatics (2010).

This paper describes a scheme for formulating non-local problems in the statics or finitr eltfinite elasticity.

EL

C. Pure Mathematics

**Description
of
Finite-Dimensional Spaces, Algebra, Geometry, and
Analysis, **

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 1987. The present version, posted on my website in 2006, 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.

I have been accused of being too "Bourbakistic". I plead guilty. I believe the work of Bourbaki was the most important contribution to mathematics in the 20'th century. Bourbaki was not a single individual but a group, some of whose members I met personally. Bourbaki was started, in 1935, by some young French mathematicians who disliked the way mathematics was taught in France at the time. To quote from the book Bourbaki, a Secret Society of Mathematicians, by Maurice Mashaal: "Gradually, the group's extensive reflections and lively discussions led to a new vision of mathematics, a modern way of teaching and even doing it."In 1973
my colleague
and friend Juan
Schaffer and I
became involved in an undergraduste honors program at CMU entitled
"Mathematical Studies". We disliked the
way mathematics was taught at the time and proposed a new way to
present mathematics as an integrated whole and to avoid its traditional
division into separate and seemingly unrelated courses, My involvement
in this program finally lead to this treatise. 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.

393 pages
(1987).

1) Finite-Dimensional Spaces, Algebra, Geometry, and Analysis Vol. I,

Introduction and 0. Basic Mathematics

Linear Spaces

2. Duality, Bilinearity

3. Flat Spaces

4. Inner-Product Spaces, Euclidean Spaces

5. Topology

6. Differential Calculus

7. Coordinate Systems

8. Spectral Theory

9.The Strucure of General Lineons

Index

2)**Finite-Dimensional Spaces, **

**Algebra,
Geometry,
and
Analysis,
Volume
II
**(1994).

** **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 .

**0.
Preliminaries, **11
pages.

**2.
Differential
Calculus II**, 17
pages.

**3.
Manifolds
in a
Euclidean
Space,
**33
pages.

**4.
Volume
Integrals,
**47
pages.

3) **The Chain Rule for Higher Derivatives**
(1995), 8 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 paper is
to derive such a formula.

4) Monoids, Boolean Algebras, Materially Ordered Sets,

by Walter Noll and Brian
Seguin (2006), 11 pages.

**
**In
this
paper,
it is shown how the concept of a

** **5 pages

5) Monoids (1992),

**
**These
are
sketchy lecture notes intended to show that the concepts of
divisibility, prime elements and prime-decompositions are most
efficiently treated in the framework of monoids. The results then apply
both to the multiplicative monoid of the natural numbers and that of
the monic polynomials over a field.

19 pages.

6) Isocategories and Tensor Functors (1992),

** **
In
this
paper, I
show how the concepts of an

7) Geometry of Differentiable Manifolds,

by Walter Noll and Sea-Mean
Chiou (preliminary
version , 1994), about 95 pages.

**
**These
lecture
notes are unusual because the topic is treated in a totally
coordinate-free manner. Included are the theory of connectors and
connections, Riemannian and symplectic manifolds, and Lie groups.
Also, a coordinate-free version of Einstein's equation of general is presented.

**Introduction
and
contents **

**Chapter
1**

**Chapter
2**

**Chapter
3**

Chapter 4

**Chapter
5**

**Bibliography**

**8) Linearly
Induced
Mappings
between
Cones of Quadratic Forms,**by
Ray
E. Artz
and
Walter Noll
(1993),
24 pages.

9) On the Conjugacy of Oerthogonal Groups (2008), 10 pages.

This paper presesnts some insights that should be of interest to anybody who ever heard of ottheogonal groups.

10) Scalar Algebras and Quaternions by Ray E Artz (2009), 36 pages

An approach based on the algebra and topology of finite-dimentional real linear spaces, part 1.