I’m searching for books that allow one to discover/experiment with mathematics by using computer environments such as Mathematica/Magma/Magma/Pari-GP, etc.
Until now, I discovered these:
If you want to learn about graph theory, there is a book called Algorithmic Graph Theory that uses Sage: “Theory and algorithms are illustrated using the Sage open source mathematics software.” This book is free. There’s also a link on that site to Explorations in Algebraic Graph Theory with Sage by Beezer and Godsil, which is also free.
If you’re interested in number theory, William Stein wrote a book called Algebraic Number Theory, a Computational Approach, which uses Sage quite a bit throughout to illustrate points. And he also wrote Modular Forms, A Computational Approach which uses Sage quite a bit throughout. These are both free. The second one you can download here.
Here’s a couple of links. I am not sure that those fulfill your request, though.
This is a course in Maple by H. Boas. It is primarily aimed at learning the basics of Maple. Nevertheless, the chapter “Slicing bread with Maple” is intended as a computer-aided lecture on higher-dimensional geometry and I find it both amusing and interesting.
‘Experiments with MATLAB’ by Cleve Moler. The title is pretty much self-explanatory: it is a book in which MATLAB is employed to work out some college-level mathematical facts. This book is a spinoff of the classic Numerical Computing with MATLAB by the same author.
As these questions appear to be acceptable around these parts, I’ll add:
Exploring Everyday Things with R and Ruby which covers “learning how to assemble, process, simulate, and analyze the available data”
Excel Scientific and Engineering Cookbook
Computations in algebraic geometry with Macaulay2
This might not be exactly what you are looking for, but The Haskell Road to Logic, Maths and Programming by Kees Doets and Jan van Eijck is a wonderful introduction to discrete mathematics.
If you wanted to learn statistics, the book SAS for Data Analysis: Intermediate Statistical Methods is a book that is about learning statistics while you are using SAS. The first few sentences of the book description on Amazon read:
This book is an integrated treatment of applied statistical methods, presented
at an intermediate level, and the SAS programming language. It serves as an
advanced introduction to SAS as well as how to use SAS for the analysis of
data arising from many different experimental and observational studies.
While there are many introductory texts on SAS programming, statistical
methods texts that solely make use of SAS as the software of choice for the
analysis of data are rare.
If you are interested in cryptography, Introduction to Cryptography with Open-Source Software uses Sage.
If you are planning to learn Classical Mechanics, you should check SICM. The idea of the authors is to benefit from the clarity and the flexibility of the Scheme programming language, in order to offer a hands-on approach. It is especially useful for beginners, since the use of a programming language helps avoiding many confusing ambiguities of the mathematical language employed in traditional books on Mechanics.
The Student’s Introduction to MATHEMATICA ®: A Handbook for Precalculus, Calculus, and Linear Algebra [Paperback]
Mathematica Cookbook
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