CRC around January 2017. I am always under a lot of pressure to perform at a high level
and so in the Summer of 2016 I decided to finally try to do something about my many notes on
analysis. I had been moving them into typed form anyway and I wanted to be able to say
I had finished them before I retired. I had just turned 65 and so it was time. I also try to
find outside agencies to fund my research when I think there is a good match. I don't pretend
to do work in some area just to get a funding nod so I don't always find a reasonable place to send
my ideas. Well, one popped up in the Summer of 2016 and by the Summer of 2017 I had
a new three year grant for my work on complex systems (brain, immune system and path planning)
as well as a big book contract for five books. So I have been busy with all of this
and I put off retirement until my grant runs out in July 2020 although I still have ideas
I am working on so I am thinking I might try to get another grant and stay longer. We'll see.
Anyway, the editor of the analysis book series says I can use my own paintings for covers
so in the Summer of 2017 over a period of about 6 weeks I painted the five I wanted to use.
They all tell a story of how I helped train new generations of cephalopods to learn mathematics.
This post I will go over the first two books in the analysis sequence.
The first one is
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| The cephalopods wanted to learn advanced mathematics and decided to contact Jim. |
and the book title is "Basic Analysis I: Functions of a Real Variable". I study basic analysis in this text.
This is the analysis a mathematics major or
a student from another discipline who needs this
background should learn in their early years.
Learning how to think in this way changes one
for life. From this point on, the reader will always know how to look carefully at
assumptions and parse their consequences.
That is a set of tools one can use for their own profit
in the future. In this text, the reader learns a lot about how
functions which map numbers to numbers behave
in the context of the usual ideas from calculus such
as limits, continuity, differentiation and integration. However, these ideas
are firmly rooted in the special properties of the numbers themselves
and we take great pains in this set of notes to place these ideas
into the wider world of mappings on sets of objects.
This is done anecdotally as we cannot study those things
in careful detail yet but we want you exposed to more general things.
The text also covers ideas from sequences of functions, series of functions
and Fourier Series.
This is a critical course in the use of abstraction and it is just one
in a sequence of courses which prepare students to become
practicing scientists. It is important to balance the theory and abstraction
with clear explanation and argument so that students who are from
many areas can follow this text and use it profitably for self study
even if they can not take this material as a course. Many professionals
need to add this sort of training to their toolkit later and
they will do that if the text is accessible.
Presentation lectures are available, along with sample exams and so forth, at
Advanced Calculus One and Advanced Calculus Two. This book is for a two semester
course.
The First Semester:
This semester is designed to cover through the consequences of
differentiation for functions of one variable and to cover
the basics leading up to extremal theory for functions of two variables.
Along the way, a lot of attention is paid to developing the theory
underlying these ideas properly. The usual introductory chapters
on the real line, basic logic and set theory are not covered as
students have seen that before and frankly are bored to see it again.
The style here is to explain very carefully with many examples
and to always leave pointers to higher level concepts that
would be covered in the other primers.
The study of convex functions and lower semicontinuity
are also introduced so that students can see
there are alternate types of smoothness that are useful.
The number e is developed and all of its properties from the
sequence definition $e = \lim_n( 1 + 1/n)^n$. Since ideas from
sequences of functions and series are not yet known, all of
the properties of the exponential and logarithm function
must be explained using limit approaches which helps tie
together all of the topics that have been discussed.
The pointer to the future here is that the exponential
and logarithm function are developed also in the second
semester by defining the logarithm as a Riemann integral
and all the properties are proven using other sorts of tools.
It is good for the students to see alternate pathways.
Note that this text eschews the development of these ideas
in a metric space setting although we do talk about metrics and norms
as appropriate. It is very important to develop
the derivative and its consequences on the real line
and while there is a simplicity and economy of expression
if convergence and so on is handled using a general metric,
the proper study of differentiation in $\Re^n$ is not as amenable
to the metric space choice of exposition, That sort
of discussion is done in a later primer.
Chapters 1 - 14 are the first semester of undergraduate analysis.
The Second Semester:
The second half of this primer is about developing the theory of
Riemann Integration and sequences and series of functions carefully.
Also, the students horizons are expanded a bit by showing them
some basic topology in two D and three D and revisiting sequential
and topological compactness in these higher dimensions. This ties
in well with the last chapter of semester one. These ideas are used
at the end to prove the pointwise convergence of the Fourier Series of a function
which is a great application of these ideas. In the text, Chapters 15 - 28 are for the second semester.
The current table of contents is given below as snapshots from the book pdf.
The next book uses this cover:
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| The cephalopods were eager to learn. Now both squid and octopi are coming to the lectures. |
and the title is "Basic Analysis II: A Modern Calculus in Many Variables". This book introduces
more ideas from multidimensional calculus.
In addition to classical approaches, I also discuss
things like one forms and
a reasonable coverage of one of the multidimensional versions
of the one variable Fundamental Theorem of Calculus ideas.
I also continues your training
in the abstract way of looking at the world. I feel that is a most important
skill to have when your life's work will involve quantitative
modeling to gain insight into the real world.
The first text essentially discusses the abstract concepts underlying the study of calculus on
the real line. A few higher dimensional concepts are touched on such as the development
of rudimentary topology in two D and three D, compactness and the tests
for extrema for functions of two variables, but that is not a proper study
of calculus concepts in two or more variables. A full discussion of the n dimenisional
based calculus is quite complex and even this second primer can not cover all the
important things. The chosen focus here is on differentiation in n dimensions
and important concepts about mappings from n dimensions to m dimensions such
as the inverse and implicit function theorem and change of variable
formulae for multidimensional integration. These topics alone require
much discussion and setup. These topics intersect nicely with
many other important applied and theoretical areas which are no longer
covered in mathematical science curricula. The knowledge here allows
a quantitatively inclined person to more properly develop multivariable
nonlinear ODE models for themselves and physicists to learn the proper background to
study differential geometry and manifolds among many other applications.
However, this course is just not taught at all anymore. It is material
that students at all levels must figure out on their own. Most of the
textbooks here are extremely terse and hard to follow as they assume
a lot of abstract sophistication from their readers. This text is designed
to be a self - study guide to this material. It is also designed
to be taught from, but at my institution, it would be very difficult to
find the requisite 10 students to register so that the course could be taught.
Students who are coming in as Master's Students generally do not have
the ability to allocate a semester course like this to learn this material.
Instead, even if they have a solid introduction from the the first course on analysis,
they typically jump to what is done in the third book, which is an introduction to
very abstract concepts such as metric spaces, normed linear spaces
and inner products spaces along with many other needed deeper ideas.
Such a transition is always problematic and the student is always
trying to catch up on the wholes in their background.
Also, many multivariable concepts and the associated theory
are used in probability, operations research and optimization
to name a few. In those courses, n dimensional based ideas
must be introduced and used despite the students not having
a solid foundational course in such things. Hence, good students
are reading about this themselves. This primer is intended to give
them a reasonable book to read and study from.
There is also provide an introduction to winding numbers for sets in two D
which uses line integrals and Green's Theorem in the plane.
New models of signals applied to a complex system use these
sorts of ideas and it is a nice way to plant such a pointer to future things.
Presentation lectures will be available for this course although not written yet.
Hence, this text would be a primary text for a senior year course that follows the first course
in basic analysis. However, this course is not taught very often now,
so it is designed for self-study and recommended as the appropriate primer
for anyone whose work requires that they begin to assimilate more
abstract mathematical concepts as part of their professional growth after graduation
or as a supplement to deficiencies in undergraduate preparation that
leaves them unprepared for the jump to the first graduate level analysis course,
which is book three.
A rough table of contents is this:
First the ideas of compactness are extended to two D and three D
along with basic ideas of convergence and continuity. Hence, the notion
of the consequences due to continuity on a compact set emerge early.
There is an extensive study of mappings from
n dimensions to m dimensions culminating in proofs of the implicit and inverse
function theorem. There is therefore a complete discussion of differentiation in n dimensions
to allow the proper development of these ideas.
There is reasonably deeper look at extremals in n dimensions which castes
extremal conditions in terms of positive definiteness and matrix minors
which is not usually discussed but which is very important.
Riemann integration is then developed in two D and higher dimensions
and characterization of Riemann integrable functions in terms
of their sets of discontinuities is included. This allows the important ideas
of sets of measure zero to be broached which is a nice pointer to
the text in measure theory in book four. The text finishes with
a good discussion of winding numbers, line integrals and
Fubini Theorem type results and change of variables
formulae.
The extensions of the Fundamental Theorem of Calculus
to two D and three D, surface integrals and the like are not discussed with the exception of Green's Theorem. In many ways, material of that nature should be covered using ideas from differential geometry and manifold theory
and more general ideas about integration.
The current table of contents is this (although still in flux):
That's all for today.
