2. What is Mathematica?
Mathematica is a general computer software system and language intended for mathematical and other applications.
3. You can use Mathematica as:
A numerical and symbolic calculator where you type in questions, and Mathematica prints out answers.
A visualization system for functions and data.
A high-level programming language in which you can create programs, large and small.
A modeling and data analysis environment.
A way to create interactive documents that mix text, animated graphics and sound with active formulas.
4. Mathematical computations can be divided into three main classes:
Numerical
Symbolic
Graphical
Mathematica handles these three classes in a unified way.
5. The simplest way to use Mathematica is like a calculator.
You can use Mathematica lile a traditional programming language such as Fortran or BASIC.
A traditional system might support perhaps 30 mathematical operations, Mathematica has over 800 built in.
In addition, while traditional systems handle only numerical computations, Mathematica also handles symbolic and graphical computations.
6. Here are some simple examples. Each one consists of a short "dialog" with Mathematica.
The text on the lines labeled
In[n] : = is what you type in;
the lines labeled
Out[n]= are what Mathematica prints back.
7. Numerical Computation
Whereas a traditional calculator or numerical computation system handles numbers only to fixed degree of precision, Mathematica can handle numbers of any precision.
In addition, Mathematica includes a full range of higher mathematical functions.
8. Mathematica can do numerical computations not only with individual numbers, but also with objects such as matrices.
Mathematica can handle numerical data, allowing you to do statistical and other analysis, as well as performing operations such as Fourier transforms, interpolation and least-square fitting.
Mathematica can do numerical operations on functions, such as numerical integration, numerical minimization, and linear programming.
9. Symbolic Computation
One major class of calculations made possible by Mathematica's symbolic computation capabilities is those involving the manipulation of algebraic formulas. Mathematica can do many kinds of algebraic operations.
It can
Expand,
Factor,
Simplify polynomials and rational expressions,
Find algebraic solutions to polynomial equations and systems of equations.
10. Mathematica can also do calculus. It can
evaluate derivatives and integrals symbolically and find symbolic solutions to ordinary differential equations
derive and manipulate power series approximations
find limits.
Standard Mathematica packages cover areas such as vector analysis and Laplace transforms.
11. Graphics
Mathematica produces both 2D and 3D graphics, as well as contour and density plots.
You can plot both functions and lists of data.
Mathematica provides many options for controlling the details of graphics output. In three dimensions, for example, you can control
Shading
Color
Lighting
surface shininess
and other parameters.
Most versions of Mathematica also support animated graphics.
12. Plot the function x(x+1)2(x-2)3 for x between -3 and 3.
This generates a two-dimensional plot of y=x(x+1)2(x-2)3
13. Mathematica incorporates a graphics language, in which you can give symbolic representations of geometrical objects using primitives such as polygons, then render the objects graphically. All graphics produced by Mathematica are in standard PostScript, and can be transferred to a wide variety of other programs.
14. The Mathematica Language
In addition to having a large number of built-in functions, Mathematica also includes a full programming language, which allows you to add your own extensions to the system.
Mathematica is a high-level programming language, in which you can write programs, both large and small. The fact that Mathematica is an interactive system means that you can run your programs as soon as you have typed them in.
15. Mathematica programs can make use of the symbolic aspects of Mathematica. They can create and manipulate arbitrary symbolic data structures. Mathematica programs themselves are also symbolic expressions, and can be combined and manipulated using standard Mathematica operations.
Mathematica supports several programming styles, including:
Procedural programming, with block structure, conditionals, iteration and recursion.
Functional programming, with pure functions, functional operators and program-structure operations.
Rule-based programming, with pattern matching and object orientation.
16.
17. Mathematica Interfaces
Many Mathematica systems are divided into two parts:
the kernel, which actually performs computations,
the front end, which handles interaction with the user.
The kernel works the same on all computers that run Mathematica.
The front end, on the other hand, is optimized for particular computers and graphical user interfaces.
18. On most computers, the front end for Mathematica supports sophisticated interactive documents called notebooks.
Notebooks consist of
text arranged in a hierarchical way,
together with graphics that can be animated,
Mathematica expressions that can be used for actual Mathematica computations.
With notebooks, you can create pedagogical and other material that both explains and performs computations.
19. Suggestions about Learning Mathematica
As with any other computer system, there are a few points that you need to get straight before you can even start using Mathematica. For example, you absolutely must know how to type your input to Mathematica.
Once you know the basics, you can begin to get a feeling for Mathematica by typing in some examples. Always be sure that you type in exactly what appears in the examples - do not change any capitalization, bracketing, etc.
20. After you have tried a few examples, you should start experimenting for yourself. Change the examples slightly, and see what happens. You should look at each piece of output carefully, and try to understand why it came out as it did.
After you have run through some simple examples, you should be ready to take the next step: learning to go through what is needed to solve a complete problem with Mathematica.
21. You will probably find it best to start by picking a specific problem to work on. Pick a problem that you understand well - preferably one whose solution you could easily reproduce by hand. Then go through each step in solving the problem, learning what you need to know about Mathematica to do it. Always be ready to experiment with simple cases, and understand the results you get with these, before going back to your original problem.
When you have built up a reasonable knowledge of the features of Mathematica, you should go back and learn about the overall structure of the Mathematica system. You can do this by systematically following a Mathematica book.
