Starting Mathematica
This is a basic introduction to Mathematica. Since Mathematica is a very powerful program
with many features only a small fraction of its capabilities will be discussed here. See
http://bartok.ucsc.edu/peter/115/math_intro/node1.html
or go to Mathematica\help. Unlike conventional computer languages Mathematica can do
symbolic manipulation, has a huge number of built-in numerical functions, can do
numerical calculations to arbitrary precision, and has powerful plotting routines which
interface directly with the results of calculations. Mathematica can also be used for
programming in styles which are quite different from those of C and fortran, but we will
not discuss programming in Mathematica here.
There are two interfaces to Mathematica: (i) a command line interface, and (ii) the
notebook interface. The same commands are given in both, but the notebook interface has
some additional features.
If one starts Mathematica in Windows by clicking on an icon one goes straight into the
notebook interface in which a separate window appears into which Mathematica commands are
typed. To execute a command in Windows, you need to hold down the Shift key as well as
pressing Enter (just pressing Enter allows you to continue entering your command on the
next line but does not execute it).
A Simple Mathematica Session
Mathematica understands the usual operators + - * / and for exponentiation. It also
understands basic constants pi and E. It also knows about a large number of mathematical
functions such as Exp[x], Sin[x], Cos[x], ArcSin[x], Log[x]
The following session illustrates some of these features:
x = 7
y = 19
x y
(x + 3) y
2^x
Exp[ Pi]
Two-Dimensional Plots
The basic plotting command is Plot[f, {x, xmin, xmax}]
which plots the function f(x) from xmin to xmax
For example,
The command Show can also be used to combine plots. Suppose we create plots of
two functions using two separate
calls to Plot, e.g.
Integration
Defining Functions
There are many functions in Mathematica but it is often very useful to be able to define
functions oneself. This is illustrated by the following example, which defines the
function . f(x) = x + x^2
Note two things about the first line:. First of all the underscore after the x, which is
called a ``blank'', stands for ``any expression'', so when the function is called, the
argument can be named anything, not necessarily x. Secondly, the use of :=, rather than =,
which means that the assignment is delayed until the function is called. In the present
example it wouldn't have made any difference if we had used =, the normal assignment
operator, but it would if the function depended on a parameter.
Series Expansions
.
Mathematica can work out series of complicated functions to very high order. The command
is Series[f, {x, x0, n}] which expands f in powers of up to n-th order, e.g. to get the
first 20 terms in the expansion of about x=0:
Three-Dimensional Plots