This website was created to publish some 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.
The entries will be updated periodically.
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,
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
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),
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.
I wrote a
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.
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.
 On the Illusion of Physical Space
 On Material Frame Indifference
 Updating The Non-Linear Field Theories of Mechanics
 The Theory of Simple Semi-Liquids, a Conceptual
Framework for Rheology
 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  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.
describe what I
consider to be
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),
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.
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:
2. Timed Eventworlds
3. Flat Eventworlds
4. Classical Spacetimes
5. Minkowskian Spacetimes
6. World Momentum
Thoughts on Thermomechanics (2008).
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 Rn-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."
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
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.
1) Finite-Dimensional Spaces, Algebra, Geometry, and Analysis Vol. I, 393 pages (1987).
Introduction and 0. Basic MathematicsI C0
2. Duality, Bilinearity
3. Flat Spaces
4. Inner-Product Spaces, Euclidean Spaces
6. Differential Calculus
7. Coordinate Systems
8. Spectral Theory
9.The Strucure of General Lineons
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, 8 pages.
IIC01. Determinants, Invariants, Covariants, 18 pages.
Calculus II, 17
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.
it is shown how the concept of a Boolean algebra can be based
on that of a monoid. Both of these concepts may be intricately
connected with the concept of an ordered set. In particular, it is
shown how Boolean algebras can be generated from what one of us (WN)
has previously called a materially ordered set, in the context
of the axiomatic foundations of continuum physics.
5) Monoids (1992), 5 pages
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.
6) Isocategories and Tensor Functors (1992), 19 pages.
In this paper, I show how the concepts of an isocategory and the corresponding concept of an isofunctor can be used to improve the conceptual infrastructure of many branches of mathematics. Isofunctors that involve the isocategory LIS of all linear isomorphism of finite-dimensional linear spaces are called tensor functors, because they can be used to clarify most uses of the term "tensor" in the literature of mathematics and physics. Of particular importance are the analytic tensor functors, which can serve to be the basis for a completely coordinate-free presentation of the theory of differentiable manifold (see 6 below).
7) Geometry of Differentiable Manifolds,
by Walter Noll and Sea-Mean Chiou (preliminary version , 1994), about 95 pages.
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
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.