Results matter, whether you are a mathematician, scientist, or engineer. The time that you spend doing tedious mathematical calculations could be spent in more productive ways. Sage is an open-source mathematical software system that helps you perform many mathematical tasks. There is no reason to compute integrals or perform algebraic manipulations by hand when software can perform these tasks more quickly and accurately (unless you are a student who is learning these procedures for the first time). Students can also benefit from mathematical software. The ability to plot functions and manipulate symbolic expressions easily can improve your understanding of mathematical concepts. Likewise, it is largely unnecessary to write your own routines for numerical mathematics in low-level languages such as FORTRAN or C++. Mathematical software systems like Sage have highly optimized functions that implement common numerical operations like integration, solving ordinary differential equations, and solving systems of equations.
Sage is a collection of nearly 100 mathematical software packages, which are listed at http://www.sagemath.org/links-components.html. When possible, existing tools are integrated into Sage, rather than duplicating their functionality. The entire collection of tools can be downloaded and installed as a binary distribution or compiled from source code. The Python language provides a unified interface to all of the packages. Python is a high-level, interpreted, object-oriented programming language that is already well established in the research community. Users can interact with Sage through an interactive command-line interface or a graphical notebook interface. Sage can also be used as a Python library or embedded in LaTeX documents. Sage is "officially" available for recent versions of OS X, Linux, Solaris, and Open Solaris. It runs on Windows with the help of a virtual machine and it can be used on other platforms, with varying degrees of support. A current list of all the available platforms can be found at http://wiki.sagemath.org/SupportedPlatforms.
The mission statement of the Sage project is:
Creating a viable, free, open source alternative to Magma, Maple, Mathematica, and Matlab.
If you are familiar with any of these commercial mathematical software systems, then you already have a good idea what Sage does. Sage offers several advantages over its commercial competitors. Sage is free, open-source software, released under the GNU Public License version 2 or higher (GPLv2+). There is no cost to download and install Sage, whether you want to put it on your personal computer, install it in a university teaching lab, or deploy it on every workstation in a company. This advantage is especially important in developing countries. The GPL license also means that Sage is free, as in "freedom." There are no restrictions on how or where you use the software, the license can never be revoked, and there is no annual maintenance fee. Another advantage is that you have access to every line of source code, so you can see how every calculation is performed, and track exactly what changes are made from one version to the next. Unlike commercial software, the bug list for Sage is public, and it can be accessed at http://trac.sagemath.org/. Users are encouraged to participate in the development of Sage by reporting and fixing bugs, and contributing new capabilities. With bugs and source code open for public review, you can have a high degree of confidence that Sage will produce correct results.
This book is written for people who are new to Sage, and perhaps new to mathematical software altogether. For this reason, the examples in the book emphasize undergraduate-level mathematics such as calculus, linear algebra, and ordinary differential equations. However, Sage is capable of performing advanced mathematics, and it has been cited in over 80 mathematical publications. A full list can be found at http://www.sagemath.org/library-publications.html. To benefit from this book, you should have some fundamental knowledge of computer programming, but the Python language will be introduced as needed throughout the book. The next chapter will take you through some examples that showcase a small subset of Sage's capabilities.
Chapter 1, What can You do with Sage? covers how Sage can be used for: making simple numerical calculations; performing symbolic calculations, solving systems of equations and ordinary differential equations; making plots in two and three dimensions; and analyzing experimental data and fitting models.
Chapter 2, Installing Sage covers how to install a binary version of Sage on Windows and install a binary version of Sage on OS X; install a binary version of Sage on GNU/Linux; compile Sage from source.
Chapter 3, Getting Started with Sage covers using the interactive shell; using the notebook interface; learning more about operators and variables; defining and using callable symbolic expressions; calling functions and making simple plots; defining your own functions; and working with objects in Sage.
Chapter 4, Introducing Python and Sage covers how to: use lists and tuples to store sequential data; iterate with loops; construct logical tests with "if" statements; read and write data files; and store heterogeneous data in dictionaries.
Chapter 5, Vectors, Matrices, and Linear Algebra covers how to create and manipulate vector and matrix objects; how Sage can take the tedious work out of linear algebra; learning about matrix methods for computing eigenvalues, inverses, and decompositions; and getting started with NumPy arrays and matrices for numerical calculations.
Chapter 6, Plotting with Sage covers how to plot functions of one variable; making various types of specialized 2D plots such as polar plots and scatter plots; using matplotlib to precisely format 2D plots and charts; and making interactive 3D plots of functions of two variables.
Chapter 7, Making Symbolic Mathematics Easy covers how to create symbolic functions and expressions, and learn to manipulate them; solve equations and systems of equations exactly, and find symbolic roots; automate calculus operations like limits, derivatives, and integrals; create infinite series and summations to approximate functions; perform Laplace transforms; and find exact solutions to ordinary differential equations.
Chapter 8, Solving Problems Numerically covers how to find the roots of an equation; compute integrals and derivatives numerically; find minima and maxima of functions; compute discrete Fourier transforms, and apply window functions; numerically solve an ordinary differential equation (ODE), and systems of ODEs; use optimization techniques to fit curves and find minima; and explore the probability tools in Sage.
Chapter 9, Learning Advanced Python Programming covers how to define your own classes; use inheritance to expand the usefulness of your classes; organize your class definitions in module files; bundle module files into packages; handle errors gracefully with exceptions; define your own exceptions for custom error handling; and use unit tests to make sure your package is working correctly.
Chapter 10, Where to go from here covers how to export equations as PNG and PDF files; export vector graphics and typeset mathematical expressions for inclusion in LaTeX documents; use LaTeX to document Sage worksheets; speed up collision detection using NumPy vector operations; create a Python script that uses Sage functionality; and create interactive graphical examples in the notebook interface.
Required:
Sage
If using Windows, VMWare Player or VirtualBox is also required.
Recommended, but not strictly necessary: LaTeX
Optional, for building Sage from source on Linux: GCC, g++, make, m4, perl, ranlib, readline, and tar
Optional, for building Sage from source on OS X: XCode
A web browser is required to use the notebook interface
If you are an engineer, scientist, mathematician, or student, this book is for you. To get the most from Sage by using the Python programming language, we'll give you the basics of the language to get you started. For this, it will be helpful if you have some experience with basic programming concepts.
In this book, you will find several headings appearing frequently.
To give clear instructions of how to complete a procedure or task, we use:
Action 1
Action 2
Action 3
Instructions often need some extra explanation so that they make sense, so they are followed with:
This heading explains the working of tasks or instructions that you have just completed.
You will also find some other learning aids in the book, including:
These are short multiple choice questions intended to help you test your own understanding.
These set practical challenges and give you ideas for experimenting with what you have learned.
You will also find a number of styles of text that distinguish between different kinds of information. Here are some examples of these styles, and an explanation of their meaning.
Code words in text are shown as follows: "We can use the help function to learn more about it."
A block of code is set as follows:
print('This is a string')
print(1.0)
print(sqrt)Any command-line input or output is written as follows:
sage: R = 250e3 sage: C = 4e-6 sage: tau = R * C sage: tau
New terms and important words are shown in bold. Words that you see on the screen, in menus or dialog boxes for example, appear in the text like this: "clicking the Next button moves you to the next screen".
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