What the rain knows

I’m not sure yet precisely how I should try working through Apostol. The last time that I did a mathematics book, it was a probability and stats text, and I ended up solving all of the exercises for which there were solutions in the back of the book, longhand, in a couple of marble composition notebooks. In each instance where I chose to use R or MS Excel to solve a problem (sometimes, I will sheepishly admit, instead of working it through by hand), I would print out the commands used and the results and tape them onto the relevant notebook page.

I used Gunnlaugur Þór Briem’s bash script to install Steve Mayer’s LatexRender plugin and can testify that it works, but am not sure how much practical it would be to use this blog as a notebook. Probably not very.

I’m thinking of reading CaF and Apostol in parallel.

Both volumes of my set are from the 32nd printing. The books themselves are simple, but definitely exude that can’t-quite-put-your-finger-on-it 1960’s math and engineering textbook charm.

From the Excerpts from the Preface to the First Edition

It is possible to combine a strong theoretical development with sound training technique; this book represents an attempt to strike a sensible balance between the two. While treating the calculus as a deductive science, the book does not neglect applications to physical problems.

The approach of this book has been suggested by the historical and philosophical development of calculus and analytic geometry. For example, integration is treated before differentiation. Although to some this may seem unusual, it is historically correct and pedagogically sound.

When I read the above, Apostol’s approach didn’t strike me as odd. I was an engineering student at a large East-Coast American state university in the mid-1990’s and I could have sworn that they began with integrals as well. Shock: I popped open my copy of Thomas and Finney’s Calculus and Analytic Geometry (8th ed.) just now and, lo and behold, they start out with derivatives.

From the Preface to the Second Edition

Differences from the first edition:

• Linear algebra added.
• Mean-value theorems and applications of calculus show up earlier in the text.
• Smaller chapters centered on a single main idea.

• A critique of Abelson and Sussman - or - Why calculating is better than scheming
• The Structure and Interpretation of the Computer Science Curriculum, a critical view of SICP and rationale for HtDP

Both are from the References section of the Wikipedia SICP article (Structure and Interpretation of Computer Programs).

Wadler’s 26-page 1987 paper/memo seems to have been scanned in from a really poor photocopy and looks like a declassified UFO crash report or really rambling kidnapper’s ransom note. Appearances don’t matter as much as content, but it was tough to read.

I’m going to divide the introduction into sections. The titles are mine, rather than the authors’, unless stated otherwise.

Warmup

• Comparison of predictable tides to unpredictable weather (forecasts good for much shorter intervals).
• Chaos is deterministic (future completely determined by past) - but small uncertainties can snowball.
• Mention of methodologies which have been designed for a precise scientific evaluation of the presence of chaotic behavior in mathematical models as well as in real phenomena and says that they can be used to estimate the “predictability horizon” of a system - mathematical/physical/time limit within which predictability is ideally possible.
• Predictability horizon for weather is not more than 2-3 weeks.

History of chaos theory

• Calculus developed by Newton and Leibniz.
• Laplace’s demon
• Uncertainty principle
• Lewis Fry Richardson’s book Weather Prediction by Numerical Process
• John von Neumann began building ENIAC (1st electronic computer) to try to put Richardson’s ideas about weather prediction into practice.
• It was found (by whom?) that Richardson hadn’t succeeded because the space and time increments used in his work had not met a computational stability criterion (Courant-Friedrichs-Lewy Criterion).
• Edward Norton Lorenz and the butterfly effect (aka “sensitive dependence on initial conditions”) mentioned.
• Near the end: determinism and predictability are not equivalent.

Also:

In other words, one of the lessons coming out of chaos theory is that the validity of the causality principle is narrowed by the uncertainty principle from one end as well as by the intrinsic instability properties of the underlying natural laws from the other end.

Notes

A Google search for “Courant-Friedrichs-Lewy Criterion” turned up only for-pay journal articles, all focused narrowly on particular applications/instances of its use. A Google Book search turned up a book that I’ve added to my Amazon wishlist, The Emergence of Numerical Weather Prediction: Richardson’s Dream (ISBN-13 978-0521857291), Richardson’s book, and the second volume of the authors other work, Fractals in the Classroom, which seems to have the same introduction, word for word, as CaF.

The edition of Richardson’s book cited in a footnote in the introduction is the 1965 Dover edition.

The Foreword to Chaos and Fractals: New Frontiers of Science was written by none other than Mitchell J. Feigenbaum.

Feigenbaum warms up with a discussion of fluid turbulence and then goes on to divide physical phenomena into three categories: linear (for which the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now.), nonlinear (some of which can be predicted by using distorted models of linear phenomena), and strongly nonlinear (which includes “chaos”).

Finished. Feigenbaum ended up used the foreward to push the Feigenbaum constants (Wolfram MathWorld: Feigenbaum Constant).

Chaos and Fractals: New Frontiers of Science was first published by Springer in 1992. I’ve got the 2nd edition (first printing, it seems), published in 2004.

The authors, in order of appearance on the cover and in the signature of the preface: Heinz-Otto Peitgen, Hartmut Jürgens, and Deitmar Saupe.

According to the above-linked Wikipedia page for Saupe, the book won the American Publishers’ Association award for Best Mathematics Book of the Year in 1992. I can’t find an award site to confirm this fact and it isn’t noted in either of the prefaces.

Preface to the 2nd edition

The first preface (to the 2nd ed.) notes the differences between the first and second editions:

• Two appendices have been removed - one by Yuval Fisher on the subject of fractal image compression because Fisher’s since published a book on the subject. They don’t detail the subject of the other appendix and I can’t find a table of contents for the 1st edition.
• There was a BASIC program at the end of each chapter of the first book and those have been replaced with a set of 10 Java applets posted online at http://www.cevis.uni-bremen.de/fractals/. CEVIS is the Center for Complex Systems and Visualization at the University of Bremen, which has been home, at one time or another, to all of the authors.

I noticed at least two typos in the preface. Apostrophes were missing when the authors talked about Yuval Fisher’s appendix or Yuval’s book (i.e. Yuval Fishers and Yuvals, respectively).

Preface to the 1st edition

The preface begins with a quote from James Gleick’s Chaos: Making a New Science. I’ve read and didn’t care for Gleick’s book.

Points of interest from this preface:

• This book is written for everyone, even without much knowledge of technical mathematics, wants to know the details of chaos theory and fractal geometry. They go on to say that it’s not a textbook, but not written in popsci style either.
• Fractals and chaos have, for students, brought math out of the realm of ancient history into the twenty-first century.
• Elements of Euclidian geometry are basc visible forms like lines, circles, and spheres, the elements of fractal geometry are algorithms.
• Fractals and modern chaos theory are linked by the fact that many of the contemporary pace-setting discoveries in their fields were only possible using computers.
• This book is a close relative of the two-volume set Fractals for the Classroom, which was published by Springer-Verlag and the National Council of Teachers of Mathematics in 1991 and 1992. [Fractals for the Classroom was written by the same three mathematicians who wrote Chaos and Fractals].
• The other appendix removed from the 2nd edition is revealed to have been a treatment of multifractal measures by Carl J.G. Evertsz and Benoit B. Mandelbrot.

• The entire book has been produced using the TEX and LaTex typesetting systems where all figures (except for the half-tone and color images) were integrated in the computer files. Even though it took countless hours of sometimes painful experimentation setting up the necessary maxcros, it must be acknowledged that this approach immensely helped to streamline the writing, editing, and printing.