Mathematica实战：通过可视化和计算解决复杂问题

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"Mathematica in Action: Problem Solving Through Visualization and Computation" 是一本由Stan Wagon编写的书籍，该书第三版介绍了如何利用Mathematica的强大功能进行数学探索和问题解决。书中重点关注了Mathematica 7的新工具，如动态操作和自适应三维绘图，帮助用户平滑过渡到新版本。作者通过各种案例研究，从基础到高级，展示了Mathematica在解决复杂数学问题时的广泛应用。本书强调Mathematica的多功能性以及跨领域技术结合所能产生的惊人结果。内容涵盖了广泛的主题，包括如何使用Mathematica发现新的数学现象。例如，书中展示如何设计一个可以让方形轮子自行车行驶的特殊道路，如何设计能钻出正方形孔的钻头，用双曲几何解释Banach-Tarski悖论，发现混沌系统的阴影轨道，以及利用强大的三维图形功能。全书强调可视化，每章都有精心制作的图形。所有Mathematica代码都包含在配套CD中，方便读者直接使用。 Stan Wagon是数学领域的知名作者，他的著作包括《计算数论课程》等，该书被美国图书馆协会评为2000年十大最佳数学书籍之一。他还撰写过多本关于Mathematica教育应用的书籍，如《VisualDSolve: 用Mathematica可视化微分方程》和《Animate Calculus: Mathematica笔记本用于实验教学》。本书适合Mathematica的新手和有经验的用户阅读，旨在提升读者的数学问题解决能力和利用Mathematica进行创新的能力。通过实际操作和丰富的视觉示例，读者可以深入理解Mathematica的功能，并将其应用于实际问题中。同时，书中的所有代码资源使得读者能够节省输入时间，更专注于理解和应用所学知识。

n++

abbreviates

n = n + 1

; similarly

n--

abbreviates

n = n - 1

n += i

denotes

n = n + i

; similarly for

n -= i

n *= i

, and

n = i



can serve as an abbreviation for

Sin  Π

and

Print  x

abbreviate

Sin@ΠD

and

Print@xD

, respectively. However, this notation should be used sparingly, as

f  x

is much less precise than

f@xD

in terms of scope. I use it sometimes when

adding, say,

Print  x

temporarily for debugging a program.

Postfix notation is sometimes useful.

expr f

is the same as

f@exprD

and is

most useful in something like

expr Timing

expr Simplify

Iterators occur often, especially in

loops or when lists are built using the

Table

command. An iterator has the form

8i, n0, n, step<

. The default step size is 1,

and an iterator of the form

8i, 200<

abbreviates

8i, 1, 200, 1<

. One can use

lists in iterators, such as

Table@Prime@iD, 8i, 810, 100, 1000<<D

, which

causes

to run through the list of three numbers.

There are three usages of the equal sign. An ordinary assignment of one value to a

variable is done via, for example,

a = 3

; one can also use

8a, b<= 83, 7<

, for

example, to handle two or more assignments. A delayed assignment is identified by

;

f := x

means that whenever

is called it will be replaced by

for the value

at the time of the call. This is most often used in the definition of functions via

dummy variables, as in

f@t_D:= t

; the underscore (blank) indicates that

is an

object that should take on the value of

expr

whenever

f@exprD

is encountered, so

f@exprD

will always turn into

expr

for any value of

expr

; in short,

can be

thought of as a dummy variable. Finally, equality is denoted by



in an equation

such as

SolveAx

+ x  7, xE

If@n  100, …D

) .

The front end has hundreds of features, and we will not go into all of them here.

But the ability to use typeset mathematics for input and output is very important, so

here is a brief introduction. For more on other aspects of the front end, such as

buttons and palettes, see [GG].

There are three main forms for mathematical expressions:

InputForm

Standard

Form

, and

TraditionalForm

InputForm

is the character-based, single-line

format that is familiar to users of early versions. For example, an integral would be

given as

Integrate

@Sin@Sqrt@xDDCos@xD, 8x, 0, 1<D

Stan

dardForm

is a typeset form that differs from traditional mathematics typesetting in that there

are no ambiguities.

0.2 Typesetting

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