Here are a few resources that I have come across. I am not in any way endorsing these as correct. I am merely listing ones I find interesting, partly so that I can browse them as I have time.

For a nice listing of free textbooks, see section 5.1.3 of our UTMOST CCLI grant proposal.

• Coursera, Udacity, and EdX offer many online courses in many subjects
• Khan Academy – a large collection of short videos that walk you through science topics, including math, from grade-school math through calculus and beyond. Many people really, really love these videos, but others have very valid reservations about them (see more and more and more about videos in teaching in general).
• Derek Owens has a lot of videos reviewing various concepts, including calculus.
• I've heard of students using CalcChat a lot.
• James Tanton has some very interesting math videos for the high school and undergraduate level.
• Institute for Mathematics and its Applications has a huge number of fascinating videos of talks. Mathematical Sciences Research Institute also has a huge number of great videos of talks.
• OpenLearning seems to offer some free courses.
• MathOverflow has another list of course videos.

• A Better Calculator is a really cool javascript graphing calculator.

• Math in Society by David Lippman, which explores topics in a very introductory way. Topics include voting, dividing things, statistics, etc.
• Fun Facts about mathematics

Here are a few free calculus textbooks.

• Precalculus, used at University of Washington. This seems like a very nice book.

• Hoffman's Calculus
• Calculus by Granville and then augmented by David Joyner. This includes lots of Sage examples for doing computations and graphs.
• Whitman Calculus
• Calculus by Gilbert Strang. This is a published book as well.
• Calculus by Benjamin Crowell
• Understanding Calculus by Faraz Hussain
• Differential Calculus by David B. Massey. I'm confused by this one. It doesn't seem really free (you can pay them to get the source to create your own customized version), but you can download a PDF and watch online video lectures. It's may just be good supplementary material for students. There is also an Integral Calculus book by the same author, and apparently a multivariable text is on its way.
• Paul's Online Math Notes: Calculus I, Calculus II, and Multivariable Calculus.
• Heiner's web page includes two links to calculus books he's written.
• Calculus: Modeling and Application by David A. Smith and Lawrence C. Moore. When you click on the word “Copyright” on this page, you learn that “If and when MAA chooses the publish these materials, they will no longer be available without charge from this public site. In the meantime, we are eager to have teachers and students use these materials freely and provide feedback on their experiences, good or bad. Such educational use is explicitly permitted in the conditions below.” Please read that page fully to see the full terms of use.
• http://www.e-booksdirectory.com/listing.php?category=4 - Lots of free calculus textbooks (including some of the ones mentioned above)

• Multivariable Calculus (in-progress PDF) by Ben Woodruff and Jason Grout. I teach from this (students also have a standard textbook for homework problems and additional explanation).
• Vector Calculus by Michael Corral
• Paul's Online Math Notes for Multivariable Calculus.
• http://tutorial.math.lamar.edu/
• Multivariable Calculus by George Cain and James Herod, used at Georgia Tech.
• http://www.math.cornell.edu/~sjamaar/classes/3210/notes.html – notes on differential forms (think: deep, advanced multivariable calculus)
• Differential Geometry by Shifrin
• The Calculus of Functions of Several Variables by Dan Sloughter
• Differential Forms by David Bachman—this is a rough draft of a set of notes that was later published as a book

• John Travis has a number of fantastic worksheets.

• Multivariable Calculus Quickstart
• Optimization: Gradients, Partial Derivatives, Lagrange Multipliers
• Introduction to Sage
• Functions: Cylinders, Parametric
• Functions: 3d, Vector, Surfaces
• Partial Derivatives
• Line Integrals; Physical Applications
• Line Integrals-Vector Fields
• Gradient Field/Directional Derivative
• Second Derivative Test
• Lagrange Multipliers

• ProblemText in Advanced Calculus by John M. Erdman. This is an “advanced calculus” book.
• Advanced Calculus by Shlomo Sternberg
• A bunch of real analysis books.

• The following OpenCourseWare courses overlap with our multivariable calculus course:
  • http://webcast.berkeley.edu/course_details_new.php?seriesid=2009-D-54296&semesterid=2009-D – Multivariable Calculus at UC Berkeley
  • http://ocw.mit.edu/OcwWeb/Mathematics/18-02Spring-2006/CourseHome/index.htm
  • http://ocw.mit.edu/OcwWeb/Mathematics/18-02Fall-2007/CourseHome/index.htm
  • http://www.maths.ox.ac.uk/courses/2009/moderations/applied-mathematics-ii/calculus-three-dimensions-and-applications/material
• http://math.furman.edu/~dcs/java/newton.html
• http://www.math.umn.edu/~garrett/qy/BadNewton.html
• A nice explanation of the product rule.
• Multivariable calculus and Matlab
• A nice overview of vector field visualization techniques.
• Cool visualization of the wind vector field, and another one.
• Helpful resources – a page of links.
• Michael Rogers has some very nice demonstrations, and some especially nice ones for multivariable calculus.
• Good Questions for understanding and testing calculus material.
• Vector Field Visualization of wind (current and animated—really cool!)
• Visualize current weather from NOAA
• Visualizing 2d and 3d vector fields using animation, along with work and flux

• http://math.bu.edu/odes/–This looks like an interesting approach to teaching differential equations, emphasizing modeling and computation.
• According to Thierry Dumont, “For the ODEs, the best and the most *modern* methods are there: http://www.unige.ch/~hairer/software.html”
• VFGEN generates the inputs to lots of different solvers. This is a decent listing of lots of different systems.
• DAEtools
• PyDSTool

• The PDE Coffee-table book - a series of 2-page spreads about various PDEs.
• Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations by Nick Trefethen

• The history of numerical analysis – a very nice website by SIAM, including some very interesting interviews with the pioneers.
• What every programmer should know about floating point

The following OpenCourseWare courses seem like they overlap with our numerical analysis course:

• http://ocw.mit.edu/OcwWeb/Mathematics/18-086Spring-2006/CourseHome/index.htm
• http://ocw.mit.edu/OcwWeb/Mathematics/18-335JFall-2006/CourseHome/index.htm
• Computational Science and Engineering

Here are a few other course outlines:

• http://persson.berkeley.edu/128A/
• Short course

• MPIR - MPIR is an active fork of the long-established GMP project, implementing multiprecision integers and rationals.
• MPFR - multiprecision correctly-rounded floating point computations.
• An overview of software which comprises an appendix in the Cheney/Kincaid numerical analysis book.
• A freely-available reference floating-point libm made by Sun
• Eigen - what looks like a very nice C++ library for doing matrix arithmetic. This library looks like it has a very nice API.

• Basic Quickstart
• Floating Point lab
• Repeated Roots
• Polynomial Approximation (drake)
• Splines and Netlib
• LAPACK
• Iterative Methods

• Tom Sederberg's book on computer graphics, including very nice explanations of splines and Bezier curves.
• Notes from a class on numerical computing with python
• http://www-users.math.umd.edu/~dlevy/books/na.pdf – A set of Numerical Analysis notes by Doron Levy at University of Maryland
• Stephen Pav's Numerical Analysis book follows roughly the treatment in Cheney and Kincaid. This book is free to modify and redistribute, and uses a smattering of Octave or Matlab in the exercises. You can also contribute through the open-source version repository.
• MPIR Documentation (algorithms chapter) contains a good overview of methods for doing computer arithmetic. You can also see the GMP documentation.
• Modern Computer Arithmetic by Richard Brent (think Brent's algorithm) and Paul Zimmermann (who helps develop MPFR). These gentlemen know what they are talking about.
• Matters Computational – This book (and accompanying software) describes a wide variety of algorithms, including combinatorial and numerical algorithms.
• Holistic Numeric Methods Institute has a large number of tutorials and videos about lots of different numerical methods topics.
• Data Analysis for Politics and Policy by Edward Tufte, a recognized authority in presenting information clearly
• Computational Mathematics
• A free online text from University of Waterloo also seems to be pretty impressive.

• Kahan's web page is an excellent resource for very interesting numerical issues. In particular, I enjoyed reading parts of How Futile are Mindless Assessments of Roundoff in Floating-Point Computation? and why Java numerical conventions hurt.
• Lenore Blum has some thoughts on two different models for dealing with numerical issues (a CS-centric one and a math-centric one) (this was published in the AMS Notices as well).
• Various problems calculators and computers have in computing mathematics.
• A nice article in the AMS Notices on Bezier curves.
• A challenge to find why a very simple floating point computation is having lots of errors.
• What every programmer should know about floating point numbers
• PHP had an infinite loop when (on 32-bit x87 FPU code) extended precision registers were used instead of normal floating point precision. The Java JVM also has a (different) infinite loop trying to handle the same floating point number.
• When is 1 equal to -128? When you use floating point arithmetic, of course!

• The following OpenCourseWare courses seem like they overlap with our modeling course:
  • http://openlearn.open.ac.uk/mod/resource/view.php?id=163003&direct=1
  • http://openlearn.open.ac.uk/course/view.php?id=2553
  • http://openlearn.open.ac.uk/mod/resource/view.php?id=163023&direct=1
  • http://openlearn.open.ac.uk/mod/resource/view.php?id=162993&direct=1
  • http://openlearn.open.ac.uk/course/view.php?id=2559
• http://www.capital.edu/21424/Computational-Studies/7111/ A huge list of modeling topics/projects for undergraduate students.
• Fantastic book by NIST on modeling

• http://www.math.yorku.ca/SCS/Gallery/ – Good and misleading graphs (look at the Ithaca Times example on http://www.math.yorku.ca/SCS/Gallery/context.html!)

• g3data can help extract the data from a scan of a graph.
• Social network modeling
• The Plus magazine has lots of fascinating modeling articles.
• A variety of simulation resources are listed at the SimPy homepage.
• SimPy – a python discrete simulation environment
  • Collectors
  • See also a textbook on using SimPy to teach.
  • An epidemiology demo also looks interesting.
  • A SimPy tutorial
• SimJS – a javascript discrete simulation environment
• Explanation of null hypothesis
• Book on optimization
• Another book on optimization algorithms, using lots of the nature-inspired algorithms.
• Pagmo is a python package implementing a number of optimization algorithms.
• Approximation theory - an in-progress book on Approximation Theory by Nick Trefethen

• Probabilistic Programming and Bayesian Methods for Hackers - based on Python and uses a lot of cool examples.

• US Census projection methodology (i.e., how are US population projections computed?)
• Vital Statistics for Iowa, including birth and death rates
• U.S. Census Migration estimates
• Leslie models:
  • http://resources.aims.ac.za/archive/2006/mosimanegape.pdf

• Brief review of history of hidden Markov models
• A more basic tutorial on HMM – I'd suggest starting here
• A tutorial from Rabiner and Juang
• Rabiner's paper, which is a bit dense mathematically if you don't have a probability background, but has a good introduction about the basic classes of problems.
• HMM in biology
• Hidden Markov model
• Some more resources about training hidden Markov models: MIT slides, an example, another resource

• Nice notes for a modeling class

• Linear Algebra by Jim Hefferon. This seems like a nice, standard book that is easy to read.
• A First Course in Linear Algebra by Rob Beezer. This is a very careful, nice book. If you have questions about the details of how something works, this is a great reference, as everything is cross-referenced to appropriate theorems, definitions, etc.
• Linear Algebra by Ben Woodruff. This is a radically different approach to a course in linear algebra compared to more standard texts. I think this approach has a lot of potential, and plan to use it in teaching soon.
• MIT Linear Algebra uses Strang's book and has a number of other helpful resources, like a glossary.
• Linear Algebra (in-progress PDF): my edited version of Ben Woodruff's linear algebra book.
• Terry Tao's online notes covering the subject look very interesting. For example, in week 3, Terry brings up nice ideas about treating bases as “units” of a vector space, so that \(\mathbb{R}\) is a quantity, but a “foot” is a basis, and one foot is just a vector with coordinate 1 in the basis of “foot” and coordinate 12 in the basis of “inch”. The concept of linear transformations also has examples in this.
• While not free, Carl Meyer's Matrix Analysis textbook is available as a pdf download.
• George Benthion has a number of notes on his site, including what looks like a very nice book on numerical linear algebra.
• Solving large-scale eigenvalue problems is a book-length set of course notes.

• Geometric explanation of Singular Value Decomposition. Also gives some of the many real-world uses of SVD.
• 19 ways to compute matrix exponentials. Co-author is Cleve Moler, the original author of Matlab)
• Visual Fourier Transformation
• Visual eigenvalues and eigenvectors

• David Arnold's linear algebra page has a lot of nice units and student projects posted, along with lots of links to ideas for student projects.
• Principal component analysis seems like another good project.

Here are some very specialized packages for computing in linear algebra.

• BLAS
• LAPACK (and cousins)
• Sage
• MATLAB
• ExpoKit - One of the most advanced software libraries for computing matrix exponentials
• Eigen - what looks like a very nice C++ library for doing matrix arithmetic. This library looks like it has a very nice API.
• http://www.netlib.org/utk/people/JackDongarra/la-sw.html A nice list of linear algebra software packages.

• Linear Algebra Quickstart
• Three class demos: Transformations, Markov models, and Unstable Inverses
• Linear Algebra Arithmetic
• Linear Algebra Applications: Vector Fields and Derivatives
• Linear Algebra Applications: Fitting and Regression
• Linear Algebra Applications: Markov Processes, Kirchoff's Law, Cramer's Law
• Linear Algebra Applications: Vector Spaces
• Linear Transformations
• Linear Algebra Applications: Discrete Cosine Transform (JPEG Compression)
• Lab 1: A matrix and its transpose
• Lab 2: Canonical Bases
• Lab 4: Discrete Cosine Transform

• http://wwwteo.informatik.uni-rostock.de/isgci/japp.html - A huge database of classes of graphs and their inclusions.

• Raazesh Sainudiin's homepage has a number of free resources (book-like lecture notes, comprehensive sets of Sage worksheets) for a number of probability-related courses.
• Distributome has a number of free javascript resources about various distributions. Another site with javascript is http://www.math.uah.edu/stat/.
• Collaborative Statistics - a popular free open-source textbook for statistics
• 40 years of boxplots - a nice introduction and history of boxplot-type plots
• http://www.openintro.org/ – Stats textbook
• Peter Norvig gives an excellent overview of common problems in experiment design.
• Zipfian has a big list of resources to learn about data analysis, many of which are free.
• Statistics done wrong – a guide to common errors in doing statistics.
• Correlation does not imply causation (Evidence). Also, Frequentist vs. Bayesian

• Physics for Entertainment
• Mechanics by Spivak
• Lots of course notes
• Stack Exchange list of *lots* of free textbooks.
• Motion Mountain–one of the longest-running free physics textbooks.

• GVSU Proofs book

• NIST Digital Library of Mathematical Functions - The reference book for all sorts of mathematical functions
• Abstract Algebra: Theory and Applications by Tom Judson
• College Algebra (CC licensed, source provided)
• Handbook of Applied Cryptography
• OpenCourseware notes on error correcting codes
• http://dimensions-math.org/Dim_E.htm - Some very nice videos talking about dimensions.
• Geometry Unbound by Kiran Kedlaya, based on notes for the Math Olympiad team.
• http://www.amazon.com/dp/0070353700 I've heard this is a good reference book, but I have never read it
• Probability and Statistics by Norman Matloff. Norman also has a number of other open textbooks (see his webpage.
• Algorithms
• Enumerative Combinatorics - a classic by Richard Stanley. The first volume is free online.

• Better Explained has some very nice blog posts on math
• University of Minnesota listing of free textbooks (professors are in the process of reviewing them too).
• Saylor – lots of college courses broken down to objectives and specific free material for each objective
• Online College Class material (lots of free books and very organized)
• Open Curriculum Resources by Rob Beezer. This is a very list of good open-source math textbooks
• http://code.google.com/u/nguyenminh2/ Minh Nguyen is working on several open-source math books.
• Curriki - A site spearheaded by Scott McNealy, co-founder and former CEO of Sun Microsystems, with lots of free textbooks.
• http://people.math.gatech.edu/~cain/textbooks/onlinebooks.html - A very good listing of various online mathematics textbooks
• http://www.e-booksdirectory.com/listing.php?category=3 - Lots of free mathematics textbooks
• http://mathbooks.110mb.com/mylist.php - Lots of free mathematics textbooks
• http://www.openbookproject.net/books/ - Lots of open-source books
• http://opened.creativecommons.org/Main_Page - Open education, hosted by Creative Commons
• http://www.oercommons.org/
• http://ocwfinder.com/ Opencourseware finder
• http://www.oerrecommender.org/
• http://ocwconsortium.org/ The OpenCourseware consortium. This is an organization of Universities that have put a lot of very good materials online for free. Universities include MIT and UC Berkeley, plus lots more in the U.S. and around the world.
• http://oyc.yale.edu/ Open Yale Courses
• http://education-portal.com/article_directory/Free_Online_Courses_and_Education.html Listing of free online courses
• http://freescience.netsons.org/ (see http://www.freescience.info/Mathematics.php for the old version of the webpage)
• http://en.wikibooks.org/wiki/Subject:Mathematics
• http://www.textbookrevolution.org/index.php/Book:Lists/Subjects/Mathematics
• http://books.slashdot.org/books/04/03/04/028253.shtml–five free calculus books
• http://opencontent.org/ - Website helping people learn about open educational resources.
• http://www.slideshare.net/rreo/overview-of-open-educational-resources-oers-faculty-presentation/1
• http://www.wikieducator.org/Main_Page
• http://cnx.org/
  • http://cnx.org/content/col10522/latest/ – Statistics book
• http://academicearth.org/
• http://discovered.creativecommons.org/search/
• http://flexknowlogy.learningfield.org/2009/02/05/7oer/ A short analysis on the remixability of several resources.
• Community College Consortium for Open Educational Resources – a huge list of resources including many free textbooks
• http://www.techbooksforfree.com/
• Free K-12 textbooks
• http://www.archive.org/details/texts has a huge collection of books online, including many mathematics textbooks. For example, a search for “Calculus” yields 996 titles.
• Machine learning (preprint is available for download as of 22 Aug 2011.
• How to turn a sphere inside out: Part I, Part II

• MacTutor Archive - This is probably one of the best web references on math history.
• National Curve Bank
• http://jeff560.tripod.com/mathword.html - When and how mathematical terms were used long ago

• Mobius transformations
• The Geometry Center videos look pretty neat
• Listing of videos with significant math content

• Enumeration and Backtracking - Pre-Fascicles of The Art of Computer Programming, Volume 4A
• A ton of free books on programming
• Structure and Interpretation of Computer Programs
• Software Carpentry–lots of lectures on various computer science subjects.
• Why C++ is defective - an interesting article looking at C++
• You need to pick the right tool (language) for the job. This site compares different uses for different languages.
• I've got to say: Lisp is beautiful, technically and aesthetically. Macros are one of the beautiful things about Lisp.

Here are some great resources for learning scientific python:

• A tutorial covering a lot of the scientific tools that you may be using
• The Python bootcamp at Berkeley in August 2012
• A collection of IPython notebooks highlighting a number of different areas.

Here are some general Python resources

• Python Module of the Week, a blog focusing on illustrating the python standard library through examples.
• Dive into Python. This is one of the best books I've found to learn Python, and is free. There are two versions, one for Python version 2.6 and another for Python version 3. Sage uses Python version 2.6 currently, but the version 3 book here has rewritten a lot of the text and may be a better book even for 2.6.
• Project Euler offers lots of python examples, viewable after you complete the task
• A Byte of Python
• Python for nonprogrammers
• Online Python Learning – a nice set of interactive tutorials
• Learn Python the Hard Way – a series of exercises guiding you through the language
• Idiomatic Python
• A highly recommended book is Mark Lutz's Programming Python or Learning Python.
• How to think like a computer scientist, Python edition (Second edition)
• A Python book tutorial
• http://rexx.com/~dkuhlman/python_book_01.html
• http://inventwithpython.com/ Beginning Python programming, oriented towards making simple games (uses Pygame for some of the book).
• http://python-history.blogspot.com/ A series of blog posts about the history of Python, straight from its creator.
• http://www.skylit.com/mathandpython.html A book with several chapters online I should take a look at sometime.
• ahh is an interesting-looking python package for doing opencl stuff.
• This page and this page provide useful resources for converting code between Matlab and Sage/Python/Numpy. There is also a Matlab to Python translator for matlab files.
• Norman Matloff also has a Python tutorial book.
• PyCharm looks like an interesting (commercial) Python IDE. It's apparently free to open-source developers working on open-source projects.
• Mailing list message 1 and Mailing list message 2 also contain helpful resources (it's easier to look at these messages in html).
• Python standard library
• Python Grimoire
• Software carpentry; python

• http://eloquentjavascript.net/

• http://sixrevisions.com/resources/git-tutorials-beginners/ has a nice listing of good introductory git tutorials.
• I find both GitX and SourceTree to be very good graphical git clients. The Github clients for Windows or Mac look very interesting too.

• I am constantly referencing The Short Math Guide For Latex.
• http://www.macrotex.net/texbooks/ - a nice collection of Tex and Latex resources.
• detexify will help you find latex symbol names based on just drawing their picture.
• Use Web Equation to convert handwriting to equations
• Online latex editors include:
  • Spandex.io
  • writeLatex
  • shareLatex
  • verbosus
  • latexlab
  • flylatex
• Latex Templates has a large number of very nice templates.

• http://www.math.tau.ac.il/~nogaa/PDFS/null2.pdf
• http://oli.cmu.edu – Carnegie Mellon Open Courses

• Computational Complexity: A modern approach by Barak and Arora was recommended to me as one of the best textbooks on computational complexity out there. The website for the book has an early draft and other supplementary material.
• A compendium of NP problems.

• Information Theory

This comment on HackerNews suggests that Rosalind is a good way to get into bioinformatics.

• http://freedomdefined.org/Licenses/NC An essay arguing why the noncommercial restriction for Creative Commons licenses is bad.
• http://opencontent.org/blog/archives/347 The problem of incompatibility of CC-by-sa and GFDL
• http://chronicle.com/blogPost/David-Wiley-The-Parable-of/7244 The crisis in publishing today. I think this article downplays the contributions publishers make (like editing, graphics, etc.) in some cases, but it also raises some very interesting issues.
• A collection of links about open-access journals and the crisis in publication.

• http://www.sensepublishers.com/catalog/files/9789087901554.pdf Learning through games
• Academic Commons focuses on liberal arts education and technology.
• Open source CS capstones focuses on giving a student real-world experience in their capstone with distributed development that makes a difference in the world.
• A student special interest group that has an emphasis in making textbooks affordable.

Mathematician: noun, someone who disavows certainty when their uncertainty set is non-empty, even if that set has measure zero.

99 instances of bugs in the code...
99 instances of bugs, ....
code one out, mark it out (without running full tests),
106 instances of bugs in the code...
106 instances of bugs in the code...
106 instances of bugs, ....

(from here)

A tongue-in-cheek history of computer languages.

Walking on water and developing software from a specification are easy if both are frozen.

• For those that read Russian, a Slashdot comment mentions “USSR science textbooks. Seriously, they are great (with some obvious exceptions :) ) and they are out of copyright. For example, Fichtenholz's “Differential and Integral Calculus” is THE best textbook on calculus ever created. It's so clear and written in so beautiful language that I had actually re-read it just for fun. I don't know if there are translations into English, alas. Landau and Lifshitz's “Course of Theoretical Physics” is the one of the best reference books for the modern physics, and it's available in English. It's out of copyright but its translations might be copyrighted.”
  • (ǀᴉɐɯə uo əɹnʇɐuɓᴉS) ¿uʍop əpᴉsdn ədʎʇ noʎ uɐɔ

Here are a few good resources for analyzing numbers and sequences.

• Online Encyclopedia of Integer Sequences
• Combinatorial Object server
• The Stony Brook Algorithm Repository
• A list of lots of tools
• There are several good calculators for guessing symbolic expressions given a numeric value, like RIES, inverse symbolic calculator (version 2; see also [[http://en.wikipedia.org/wiki/Inverse_Symbolic_Calculator|Wikipedia).

Here are some interesting-looking ipod apps that I might try someday.

• http://www.spacetime.us/iphone/
• http://itunes.apple.com/us/app/mycalculator-pro-plot-graphs/id299460923?mt=8
• http://www.iphone-calc.com/wp/
• http://www.openstackinc.com/iCalc3D/
• http://www.mpkju.fr/~touchplot/
• http://pocketcas.com/iphone/
• http://itunes.apple.com/app/imatrixlab/id389947262?mt=8
• http://itunes.apple.com/us/app/id333254574?mt=8
• http://download.cnet.com/Quick-Math-CAS-Calculator/3000-20415_4-75168765.html

• Circos for drawing circular graphs
• Hive plots for drawing complex networks
• Colorbrewer for picking nice color schemes