Welcome to Math for Sustainability

This is the support website for students and instructors using the book Mathematics for Sustainability (Springer, 2018).  This website is maintained by the authors and is not in any way managed by Springer-Verlag.

Mathematics for Sustainability is a text for a new kind of college math course.  First of all, it is designed from the ground up for “non math” people. Such folk regularly have to take three credits of math to fulfill a “general education” or “distribution” requirement.  The idea of this book is to build on this requirement and design a course that will help you as a world citizen in the 21st century.  Are we going to bequeath to our children a planet that’s in better shape than the one we inherited from our parents?  That is the question of sustainability, in a nutshell, and  to think wisely about it you need some numbers and figures – you need a bit of math.  The book is intended to provide that.

Here are some ways that you can use this website:

  • Interactive software that can be used with the book can be found here.  This allows you to view dynamic (changing) versions of models in the book that we’ve only been able to indicate by fixed diagrams.
  • Material about student writing can be found here.  Extended writing has been an important component of the course and the resources we provide suggest one way to manage this.
  • Additional case studies, supplementing those in Chapter 7 of the book, can be found here.  There are many ways of applying the ideas in Mathematics for Sustainability and we only had space in the book for a few examples.  We’ll continue growing this section as we learn of new examples.  If you have an idea for a case study, please let us know!
  • Corrections to the book can be found here.  If you think you’ve found an error, or if you have a suggestion for improvement, please email us at comments@math-for-sustainability.com  We’ll keep the list up-to-date and aim to incorporate your comments and suggestions in the next edition.
  • Hints and solutions to selected exercises can be found here.
  • Author responses to your questions can be found here.  If you have a question about any of the material in the book, feel free to shoot us an email at comments@math-for-sustainability.com – we’ll post your question (anonymously) together with some thoughts about it.
  • Suggestions for instructors in the course can be found here.

All of these items are collections of blog posts which allow their own comments through the WordPress comment mechanism, if you wish.  Mathematical writing through MathJax is supported  (this page gives an introduction to how to use MathJax).

Enjoy the book! We are talking about serious stuff here, but we hope that you’ll also find pleasure in the unexpected power of mathematics.  Especially, perhaps, if you think you’re “not a math person”.

The Betz limit

There is an interesting discussion to be had about the possible efficiency of wind turbines which presents opportunities for the instructor to look at several different kinds of computations.  The discussion ends up with an optimization problem which could be approached by the classical methods of calculus – if the students have that available to them – or could be approximated simply by drawing a graph of one critical function. Continue reading “The Betz limit”

Sample question and response

A reader asked “Why did the chicken cross the road?”

It is well known that this is a hard question to analyze rigorously.  However, by starting with the 1-dimensional chicken diffusion equation

\[ \frac{\partial^2\chi}{\partial x^2} + \frac{1}{4\sqrt{\pi}}\frac{\partial \chi}{\partial t} \approx 0, \]

where \(\chi\) is the local density of chickens, we may arrive at the answer, “Because it wanted to get to the other side (to a first order approximation).”

Page 336 Sea Ice Data

A couple of exercises for chapter 5 refer to a dataset of Arctic sea ice extent, and we also plan to add an online case study on regression to the mean where this will be one of the examples.  The data comes from the National Snow and Ice Data Center Arctic Sea Ice News and Analysis.   The dataset is more completely summarized in the graphic below.

Sea Ice Extent, from Arctic Sea Ice News
All NSIDC data is available for download here.

Page 315 Extreme value calculator

This post is a supplement to the discussion of extreme value statistics at the end of Section 5.3 of the book.  You can find an online extreme value distribution calculator, provided by South Dakota State University, at


This calculator fits a Gumbel distribution (a form of generalized extreme value distribution) to a data set.  It uses the language of river floods because that is what the authors are interested in, but the underlying mathematics applies to many different situations.

To use the calculator one provides a data series consisting of extreme values.  For instance, one might provide the data series

12.1; 11; 2; 1.8; 16.4; 6.7; 8; 3;  4; 9;

which represents the biggest value of some variable (the depth of the deepest flood, the windspeed of the strongest storm, or whatever) in each of 10 successive years.  The output of the calculator is a table giving a probability distribution.  It has five columns: the key ones are labeled T (return period), P (probability) and Q (“flood discharge” for this calculator, but it refers to whatever variable we are modeling).  Here is part of the output for the data series above:


Return period T, year Probability P, percent Value Q
25 4 21
50 2 25
100 1 28

This tells us (based on the data provided) that, for instance, the value \(Q=28\) will be exceeded only once in a hundred years; the value \(Q=25\) will be exceeded only once in fifty years; and so on.