探索Wolfram语言：Mathematica在现实世界的應用

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"Mathematica Beyond Mathematics. The Wolfram Language in the Real World" 是一本由José Guillermo Sánchez León撰写、CRC Press出版的书籍，它着重介绍了Mathematica（现称为Wolfram语言）的新功能，并通过实际案例来展示这些功能在现实世界中的应用。书中涵盖了一系列广泛的话题，如能源利用、制造过程中的公差计算、城市热岛效应、文学作品与数学的关联以及太阳系外行星的寻找等，展示了Wolfram语言的强大解决能力。 Wolfram语言是一种强大的计算语言，由Wolfram Research公司开发，它集成了数学、科学、工程、人文等多个领域的知识和算法。书中的"Beyond Mathematics"部分强调了Wolfram语言不仅仅局限于数学问题的解决，还可以用于处理各种现实世界中的复杂问题。例如，通过Wolfram语言，作者可能探讨了全球能源消耗的来源，分析了制造业中的公差计算方法，以了解其对产品质量和生产效率的影响。此外，书中可能还讨论了城市热岛效应，即城市区域比周边乡村地区温度升高的现象，以及如何使用Wolfram语言进行数据分析以验证这一现象。对于文学爱好者和数学爱好者来说，书中可能包含了一个有趣的例子，即分析了《堂吉诃德》这部小说是否与圆周率π有关联。这是一个将文学与数学结合的示例，展示了Wolfram语言在文本分析和模式识别方面的应用。此外，书中可能还介绍了如何利用Wolfram语言的计算和数据处理能力，来搜索和分析天文学数据，以发现太阳系外的行星。这本书不仅适合已经熟悉Mathematica的用户，也适合希望学习如何运用高级计算工具解决现实问题的读者。通过深入浅出的实例，读者可以学习到如何利用Wolfram语言的最新特性，来处理日常工作中遇到的各种挑战，无论是科学研究、工程设计还是数据分析。书中的每个例子都旨在激发读者探索更多可能性，提高使用Wolfram语言解决实际问题的能力。

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◼ We find the intersection points of a circumference and a parabola and graphically represent

them. To write special symbols such as “∧” , the best thing to do is to use Palettes, which

we will discuss later on.

pts =Solvex

+ y

1 ∧ y - 2 x

0, {x, y}

x →

5 

5  , y →

1 

5 , x →

5 

5  , y →

1 

5 ,

x →

5 +

5  , y →



5 1, x →

5 +

5  , y →



5 1

Show

ContourPlotx

⩵1, y -2x

⩵0, {x, -1.5, 1.5}, {y, -1.5, 1.5},

Graphics[{Red, PointSize[Medium], Point[{x, y}/. pts]}]

Apart from the examples given above showcasing some of Mathematica’s new functionality, it

is also worth mentioning: (i) The Wolfram Predictive Interface, that makes suggestions about

how to proceed after executing a command, (ii) The context-sensitive input assistant to help us

choose the correct function, (iii) The possibility of running the program from the browser

(WolframCloud.com), including specific cloud-computing related commands, (iv) The

machine learning capabilities that, for example, enable us to analyze a handwritten text and

keep on improving its translation through repetition, (v) Very powerful functions for graphics

and image processing. We will refer to these and other new capabilities later on.

1.2 First Steps

The program can be run locally or in the cloud accessing the Wolfram Cloud

(http://www.wolframcloud.com/) with a browser. From here on, except mentioned otherwise, it

will be assumed that you have installed Mathematica locally and activated its license. When

installing it, an assistant will guide you through the activation process. If this is the first time, it

will take you to the user portal (https://user.wolfram.com), where you will have to sign up. The

website will ask you to create a Wolfram ID, use your email address, and your own password.

It will also automatically generate an activation key. Remember your Wolfram ID and

password because they will be useful later on if, for example, you would like to install the

program in a different computer or have an expired license. In both cases, you will need a new

activation key. The Wolfram ID and password will also be necessary to access the Wolfram

4 Mathematica beyond mathematics

Cloud, and we will give a specific example at the end of the chapter.

To start the program in Windows, under the programs menu in your system, click on the

Mathematica icon. Normally there are two icons; choose Mathematica and not Mathematica

Kernel.

Under OS X, the program is located in the applications folder and you can launch it either

locating it with Finder, using the Dock or from the launchpad (OS X Lion or more recent

versions).

By default, a welcome screen (Figure 1.1) similar to this one will appear:

Figure 1.1 The Welcome Screen in Mathematica 11.

This screen contains several external links (hyperlinks). For example, by clicking on

Resources, you will be able to access a broad collection of reference materials about how to

use Mathematica, including videos: http://www.wolfram.com/broadcast/. You can always go

back to this screen from the menu bar: Help ▶ Welcome Screen... . In the upper left-hand

corner (in Mathematica 10/11) you will see this icon:

. If you are a registered

user, chances are you already have a Wolfram ID. In that case, click on the icon and sign in.

Once the process has been completed, the icon will be replaced with the name that you used

when you signed up, as shown in the previous image. The advantage of this is to gain access

from within Mathematica to your files in the cloud (WolframCloud) but for now, you can skip

this step if desired. If you click on NewDocument, below the icon shown at the beginning of

the paragraph, a new blank Notebook will be created. Notebooks are the fundamental way to

interact with Mathematica. You can also create them by clicking on File ▶ New ▶

Notebook.nb.

A notebook is initially a blank page. We can write on it the text and instructions that we want.

It is always a good idea to save the file before we start working on it and, as in most programs,

this can be done in the menu bar: File ▶ Save As ▶ “Name of the File” (Mathematica by

default will assign it the extension .nb).

It is always convenient to have access to the toolbar from the beginning of the session; to do

that, select in the menu bar Window and check Toolbar ▶ Formatting (or Show Toolbar in

Version 9 and earlier). Please note the drop-down box to the left of the toolbar (Figure 1.2),

that we’ll refer to as: style box. It will be useful later on.

Figure 1.2 Mathematica’s Formatting Toolbar.

Once you begin to type, a new cell will be created. It will be identified by a blue right-bracket

square (

]) (“cell marker”) that appears on the right side of the notebook. Each cell constitutes a

1. Getting Started 5

unit with its own properties (format, evaluable cell, etc.). To see the cell type, place the cursor

inside or on the cell marker. The type will appear in the style box (Figure, Input, Text, Title,

Program...). We can also change the format directly inside the box. Type in a text cell and try

to change its style.

ThisisacellformattedusingtheProgramstyle.

When a blank notebook is created, it has a style (cell types, fonts, etc.) that assigns to each type

of cell certain properties (evaluable, editable, etc). The first time a notebook is created,

Mathematica assigns a style named Default (by default). Later on we will see how to choose

different styles. In the Default style, when a new cell is created, it will be of the Input type,

which is the one used normally for calculations. When we evaluate an Input type cell, “In[n]:=

our calculation” will be shown and a new cell of type Output will be generated where you will

see the calculation result in the form of “Out[n]:= result” (n showing the evaluation order).

However, in this book we sometimes use an option to omit the symbols “In[n]:=” and

“Out[n]:=”. These and other options to customize the program can be accessed through Edit ▶

Preferences ....

To execute an Input cell select the cell (placing the cursor on the cell marker) and press

+ or  on the numeric keypad. You can also access the contextual menu by

right-clicking with the mouse and selecting “Evaluate Cell”.

2 +2

Figure 1.3 The Suggestions Bar.

Since Mathematica 9, when a cell is executed, a toolbar appears below its output as shown

above (Figure 1.3). This bar, named the Suggestions Bar, provides immediate access to

possible next steps optimized for your results. For example, If you click on range you will see

that a new input is being generated and a new suggestions bar will appear right below the

result:

Range[4]

{1, 2, 3, 4}

The Suggestions Bar is part of the predictive interface. It tries to guide you with suggestions

to simplify entries and ideas for further calculations. It is one more step to reduce the time it

takes to learn how to use the program.

All written instructions inside a cell will be executed sequentially. If at the end of an

instruction we type semicolon ";", the instruction will be executed but its output will not be

shown. This is useful to hide intermediate results. In an evaluation cell (Input) we can include

one or more comments, writing them in the following format: (* comment *). Nevertheless, it

is recommended to include the comments in separate cells in text format. Remember that you

can do that by choosing Text in the style box.

To create a new cell we just need to place the cursor below the cell where we are and then start

writing. We can also create a new cell where the cursor is located by pressing +.

To facilitate writing, Mathematica provides Palettes. These can be loaded by clicking on

Palettes in the menu bar. There are several palettes to make it easier to type mathematical

6 Mathematica beyond mathematics

symbols such as the Basic Math Assistant palette or Other ▶ Basic Math Input. It would be

useful from now on to keep one of them open.

Alternatively, you can use keyboard shortcuts to write special symbols, indexes, subindexes,

etc. The most useful ones are: subindex  + _ , superindex  + ^ or  + 6, fraction 

+ / and the square root symbol  + 2. Depending on your keyboard configuration, to use

some of the previous symbols you may have to press first +  (key CAPS) for example:

superindex  +  + ^.

◼ Using a palette or the keyboard shortcuts write and execute the following expression. Please

remember: an empty space is the equivalent of a multiplication symbol: that is, instead of

4×5 you can write 4 5.

4 ×5

4 -3

3

The previous Input cell was written in the standard format (StandardForm). It is the one used

by default in Input cells. To check it, position the cursor on the cell marker and in the menu bar

choose: Cell ▶ Convert To ▶. StandardForm will appear with a check mark in front of it.

Expressions can also be written without the need for special symbols by using the form:

InputForm. It is very similar to the one used by other programming languages such as C,

FORTRAN or BASIC. For instance: multiplication = " * ", division = " / ", addition = " + ",

subtraction " = " , exponentiation =" ^ ", and square root of a = "Sqrt[a]". We can replace

the multiplication symbol by an empty space. It is very important to keep that in mind since

Mathematica interprets differently "a2" and "a 2": "a2" is a unique symbol and "a 2" is

a*2.

◼ In InputForm the previous expression can be written as:

(4*5)/5 + Sqrt[4]-3^2

3

Initially, it was its symbolic calculation capabilities that made Mathematica popular in the

academic world.

◼ As an example, write and execute the following in the standard form in an input cell (use a

palette to help you enter the content).

∂

x,y

Sinx

y 

cosx y 



x sinx

y 

A customized style, to which we will refer later, has been defined in this notebook so that the

outputs are shown in a format that corresponds to traditional mathematical notation. In

Mathematica it is called TraditionalForm. We could have defined a style so that the inputs also

use the traditional notation as well, but in practice it is not recommended. However, you can

convert a cell to the TraditionalForm style anytime you want.

◼ Copy the input cell from above to a new cell and in the menu bar select Cell ▶ Convert To

▶ TraditionalForm. Notice that the cell bracket appears with a small vertical dashed line.

You will be able to use the same menu to convert to other formats.

1. Getting Started 7

∂

sinx y 

∂

x ∂ y

cosx y 



x sinx

y 

For Windows, since Windows 7, you can use the Math Handwriting Input, available under

accessories, to write the entries manually using a tablet or an electronic board.

The linguistic or free-form format (available since Mathematica 8) allows us to write in plain

English the operation we want to execute or the problem we want to solve. To start using it

press = at the beginning of the cell (input type) and you will notice that

will appear. After

that write your desired operation. Your instruction will automatically be converted into

standard Mathematica notation, and after that the answer will be displayed. Let’s take a look at

some examples.

◼ To add from 1 to 10 you can write “sum 1 to 10” or other similar English expressions such

as “sum from 1 to 10”.

sum 1 to 10

Sum

Sum[i, {i, 1, 10}]



If you follow the previous steps, the entry proposed may be different from the one shown here.

The reason behind this is that when using the free-form format, the program connects to

Wolfram|Alpha, Wolfram’s computation knowledge engine, that is constantly evolving.

Move the cursor over Sum[i,{i, 1, 10}]and the message “Replace cell with this input”

will appear. If you click on the symbol

+ located in the upper right-hand corner, additional

information about the executed operation will be shown.

If you have received any message related to connection problems, it is because the free-form

format and the use of data collections that we will refer to later on, requires that your computer

is connected to the Internet. If it is and you still have connectivity problems it may be related

to the firewall configuration in your network. In that case you may have to configure the proxy

(see http://support.wolfram.com/kb/12420) and ask for assistance from your network

administrator.

When you don’t know the correct function syntax, you can use the free-form format to

write what you want to do. Mathematica probably will show you the correct syntax

immediately, click on it and only the input in the correct notation will remain with the free-

form format entry disappearing.

The free-form input format generally works well when it’s dealing with brief instructions in

English. In practice it is very useful when one doesn’t know the specific syntax to perform

certain operations in Mathematica.

Many more examples can be found in:

http://www.wolfram.com/mathematica/new-in-8/free-form-linguistic-input/basic-

examples.html.

8 Mathematica beyond mathematics

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