物理科学与工程数学：Maple和Mathematica符号计算应用详解

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"《数学物理与工程学：符号计算应用手册》是一本针对物理科学专业的完整数学教材，强调了符号计算在概念讲解和实际问题解决中的重要作用。该书不仅介绍了基础数学概念，还特别关注Maple和Mathematica两种符号语言的运用，帮助读者将数学知识与实际工作相结合。作者弗兰克·H·哈里斯通过深入浅出的方式，确保每个关键概念都能通过实例和插图得到清晰阐述，使学生易于理解。书中提供了丰富的学习资源，如多个附录，其中包括常用应用软件（如Mathematica和Maple）术语的概述，方便学生快速查阅和参考。符号计算在本书中被展示为解决广泛的实际问题的强大工具，无论是理论研究还是工程设计，都能通过这种方法来优化计算过程和提升效率。目标是帮助学生系统地掌握数学知识，并建立起一个在科学或工程职业中能够迅速应用的数学工具箱。该书强调实践导向，鼓励学生将所学应用于实际场景，以提高解决问题的能力，从而适应现代社会对数学技能日益增长的需求。版权方面，所有复制、存储或传输内容均需事先获得Elsevier出版社的书面许可，读者可通过电话、邮件或网站直接申请权限。《数学物理与工程学：符号计算应用》是一本既严谨又实用的教材，对于物理科学与工程领域的专业人士和学生来说，它不仅是学术研究的必备参考资料，也是提升专业技能的重要桥梁。"

1.2. BASICS OF SYMBOLIC COMPUTING 9

In this book we make extensive use of maple’s ability to draw graphs. The

plotting system gives the user the capability to select or adjust many features of

a function plot (for both ordinary two-dimensional plots and for representations of

three-dimensional objects). It requires an extremely complicated set of options to

provide this degree of control; we present here only the most basic plotting features.

The techniques described here are adequate for making a plot in which function values

are plotted against its argument, but there are other ways of generating plots, several

of which are discussed in some detail in Appendix A. That appendix also includes

a review of some of the more useful plotting options. The types of plots reviewed

in the appendix include those for functions described parametrically, contour plots,

and plots of discrete point sets. It is also shown how to combine plots of the same or

diﬀerent types in a single graph.

Two elementary 2-D plotting commands for the graphing of f(x) as a function of

x for the horizontal range a ≤ x ≤ b are

> plot(f(x), x = a .. b); Use here =,not:=.

> plot(f(x), x = a .. b, c .. d);

The second of these commands limits the vertical range to [c, d]; if the vertical range is

not given, maple chooses it to be suﬃcient to display all, or almost all, the relevant

values of f (x). If only one range is given, maple assumes it to be the horizontal

range. The horizontal range can sometimes be replaced by giving the name of the

independent variable, in which case maple chooses a default range. The quantity

being plotted, f(x), must evaluate to a number for all relevant values of x, but the

variable x must be undeﬁned.

Sometimes we want maple to print material other than a statement’s result;

examples are (1) to attach a label or explanatory matter to a numerical value, or

(2) to make a table of function values. A command to accomplish this is print.

One can print values of maple variables or arbitrary strings of characters (placed

within opening and closing double quotes). A string is not a maple variable, though

a maple variable can have a string as its value. We illustrate:

> A := "The result is";

A := "The result is"

> x := 6: print(A,x);

"The result is",6

> "The result is" := 6;

Error, invalid left hand side of assignment

Detailed control of printing is possible and is discussed brieﬂy in Section 1.5. That

section includes examples that show how to print tables of numerical data.

Many of the features discussed above are illustrated in the worksheet depicted in

Table 1.1. To save space we have placed the output of each statement to its right

instead

of centered below it on a new line. Procedures for simpliﬁcation or conversion

between equivalent forms are illustrated in Table 1.2.

maple provides an extensive run-time help facility. The command ?topic (with

no following ; or :) accesses the help page for topic. One easily ﬁnds in this way

the names and deﬁnitions of special functions and the properties and syntax of any

maple command.

It is useful to annotate worksheets with comments that inform actual or potential

users as to what is going on. In addition to the possibility of using the sharp character

1.2. BASICS OF SYMBOLIC COMPUTING 13

be numbers or they may be represented by symbols that have been predeﬁned by

mathematica and cannot be changed (e.g., Pi, which is 3.14159 ···).

The arithmetic operations are illustrated by a+b, a-b, abor a*b (multiplication),

a/b, a^b (a

). The for tran notation for exponentiation, a**b,isnot recognized by

mathematica. Multiplication and division are left-associative, so a/b*c stands for

(a/b)c and a/b/c is (a/b)/c = a/bc. When no operator symbol is used to indicate

multiplication, symbolic operands (e.g., a and b) must be separated by at least one

space; otherwise they would be interpreted as a single symbol (e.g., ab). A better

practice, followed in most of the coding examples in this book, is to always use the

* symbol for multiplication. Parentheses can be used to control grouping, and must

be used when the grouping is not that obtained in mathematica by default. For

example, a^b^c is interpreted by mathematica as a

; if the user wants (a

)

, he/she must write (a^b)^c. Argument lists for functions or commands are not

surrounded by parentheses, but by square brackets, []. Other grouping symbols or

combinations thereof, such as {}, have special meanings and cannot be used in place

of ()or [].

Commands are ended with Shift/Enter (keep the shift key depressed while

pressing Enter). Enter (by itself) simply permits a continuation of the input on the

next line of the notebook. If ; is the last character before Shift/Enter, printing

of the output (but not its generation) is suppressed. A command can consist of

just an expression, or it may be an assignment, in which case a variable is as-

signed a value (which may itself be symbolic, depending on other variables). The

assignment operator is = and once a variable has been assigned, its occurrence in all

subsequently processed statements (no matter where they physically occur in the

notebook) will be replaced by the assigned value. All output that has been generated

in a mathematica session remains available for the duration of the session, and may

be referred to as % (most recently produced output), %% (next most recent output),

%%...% (k times) (the kth previous output), or %n (the output line Out[n]). Using %n

may cause problems if a notebook is saved and reused, because the output numbering

depends on the session history. Here are a few lines of mathematica input and

output. The user does not type In[..]:=; that is supplied by mathematica when

the statement is processed.

In[1]:= x1 = z + 5

Out[1]= z +5

In[2]:= x1

Out[2]= z +5

In[3]:= % + ymax + 3

Out[3]= ymax + z +8

In[4]:= Sin[Pi/2]

Out[4]= 1

Assigns z + 5 as value of x1

Checks that assignment was

done

% denotes Out[2];

addends are combined

Note style: S and []

mathematica

knows the value

Notice that if a symbol has already been deﬁned when a statement is executed, the

value is used when the result is evaluated. Undeﬁned quantities remain as symbols.

To undeﬁne a symbol x to which something has been assigned, write Clear[x].

A need that often arises is to evaluate an expression for some value of a variable

contained in the expression. For example, we may need to evaluate x1=z+5 for z =1.

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