苏珊·简·科利尔第四版向量微积分教程

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《向量微积分（Pearson 2012 第四版，苏珊·简·科利）》是一本专为向量微积分或高等数学第三部分教学设计的专业教材。该书由苏珊·简·科利撰写，她来自奥柏林学院，内容涵盖了向量分析的基本概念、理论以及应用，适用于大学数学课程中的高级学生和工程师。本书共分为四个章节，详细探讨了向量场、多元函数微分、线性变换、格林公式、斯托克斯定理、高斯定律等核心主题，这些都是微积分中的重要分支，对于理解物理学、工程学、经济学等领域的问题解决至关重要。在第四版中，作者对前一版进行了修订和扩展，以反映最新的研究成果和技术发展，确保内容的精确性和时效性。书中配以丰富的实例和习题，帮助读者深化对向量积分的理解，并通过练习提升问题求解能力。此外，还包含了大量的图解和图表，以便于读者直观地掌握抽象概念。编辑团队由多位专业人士组成，包括首席编辑德里尔·林奇、资深收购编辑威廉·霍夫曼、赞助编辑卡罗琳·塞拉诺等，他们共同确保了书籍的质量和出版流程的顺利进行。版权和许可方面也有专门的顾问提供支持，保证了内容的合法使用。封面设计由桑苏·达达设计，插图则由亚历山德罗·德拉贝拉提供，整体风格既专业又吸引人。《向量微积分（Pearson 2012 第四版）》是学习者在深入探索数学之美的同时，提升技术分析和解决问题技巧的重要参考资料。无论是在课堂上作为教材，还是自学者查阅，它都是一个不可或缺的学习工具。

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To the Student:

Some Preliminary

Notation

Here are the ideas that you need to keep in mind as you read this book and learn

vector calculus.

Given two sets A and B, I assume that you are familiar with the notation

A ∪ B for the union of A and B—those elements that are in either A or B (or

both):

A ∪ B ={x |x ∈ A or x ∈ B}.

Similarly, A ∩ B is used to denote the intersection of A and B—those elements

that are in both A and B:

A ∩ B ={x |x ∈ A and x ∈ B}.

The notation A ⊆ B,orA ⊂ B, indicates that A is a subset of B (possibly empty

or equal to B).

One-dimensional space (also called the real line or R) is just a straight line.

We put real number coordinates on this line by placing negative numbers on the

left and positive numbers on the right. (See Figure 1.)

0123−3 −2 −1

Figure 1 The coordinate line R.

Two-dimensional space, denoted R

, is the familiar Cartesian plane. If we

construct two perpendicular lines (the x- and y-coordinate axes), set the origin

as the point of intersection of the axes, and establish numerical scales on these

lines, then we may locate a point in R

by giving an ordered pair of numbers (x, y),

the coordinates of the point. Note that the coordinate axes divide the plane into

four quadrants. (See Figure 2.)

, y

)

Figure 2 The coordinate plane R

Three-dimensional space, denoted R

, requires three mutually perpendicular

coordinate axes (called the x-, y- and z-axes) that meet in a single point (called

the origin) in order to locate an arbitrary point. Analogous to the case of R

,ifwe

establish scales on the axes, then we can locate a point in R

by giving an ordered

triple of numbers ( x , y, z). The coordinate axes divide three-dimensional space

into eight octants. It takes some practice to get your sense of perspective correct

when sketching points in R

. (See Figure 3.) Sometimes we draw the coordinate

axes in R

in different orientations in order to get a better view of things. However,

we always maintain the axes in a right-handed conﬁguration. This means that

if you curl the ﬁngers of your right hand from the positive x-axis to the positive

y-axis, then your thumb will point along the positive z-axis. (See Figure 4.)

Although you need to recall particular techniques and methods from the

calculus you have already learned, here are some of the more important concepts

to keep in mind: Given a function f ( x ), the derivative f



(x) is the limit (if it exists)

of the difference quotient of the function:



(x) = lim

h→0

f (x + h) − f (x)

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