Schoolcraft College Michael Corral's Vector Calculus: A One-Semester Guide

需积分: 25 134 浏览量更新于2024-07-18 收藏 2.48MB PDF 举报

"Vector Calculus" 是一本专门针对二元和三维空间中的微积分教学的教材，适合用于一学期的课程学习，通常称为“向量微积分”、“多元微积分”或简称为“高等数学III”。该课程的基础是单变量微积分（即微积分I和II）。本书由Michael Corral撰写，他本人是Schoolcraft College数学系的兼职教授。 Corral 拥有加州大学伯克利分校的数学学士学位以及密歇根大学的数学硕士学位和工业与运筹工程硕士学位。他的学术背景确保了本书内容的严谨性和实用性。教材采用LaTeX2ε排版，配合KOMA-Script bundle，利用GNUEmacs文本编辑器在Fedora Linux系统上完成制作。图形部分则借助MetaPost、PGF和Gnuplot工具创建，体现了作者对技术细节的把控。本书涵盖了向量代数的基本概念，如向量的定义、加法、标量积、向量积等，以及向量微分和积分，包括梯度、散度和旋度的计算。这些是理解多元函数性质、场的物理意义和偏微分方程的基础。章节可能包括多元函数的极限和连续性、多元微分规则、多元函数的极值问题、曲线和曲面的参数化、多元积分的计算方法（如格林公式、斯托克斯公式和高斯定理）等。此外，书中还会涉及黎曼球面和多维坐标系的概念，这对于理解和应用在物理学、工程学、经济学等领域的问题至关重要。作者强调实践应用，可能包含例题解析和习题，帮助学生巩固理论知识并提升解题能力。版权方面，这本书遵循GNU自由文档许可证（Version 1.2或更高版本），允许读者在一定条件下复制、分发和修改内容。附录中包含了详细的许可条款。《向量微积分》是一本既注重理论深度又兼顾实践操作的教材，对于希望进一步探索多元数学的学生来说，是不可或缺的学习资源。通过阅读和学习本书，学生可以建立起坚实的向量分析基础，为后续更高级的数学课程打下坚实的基础。

8 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE

Example 1.3. Calculate the following:

(a) Th e magnitude of the vector

−−→

PQ in R

with P =(−1,2) and Q =(5, 5).

Solution: By formula (1.2),

−−→

(5 −(−1))

+(5 −2)

36 +9 =

45 =3

(b) Th e magnitude of the vector v =(8, 3) in R

Solution: By formula (1.3), kvk=

73.

Solution: By formula (1.4), the distance d =

(4 −2)

+(2 −(−1))

+(−3 −4)

4 +9 +49 =

62.

(d) The magnitude of the vector v =(5,8,−2) in R

Solution: By formula (1.5), kvk=

+(−2)

25 +64 +4 =

93.

Exercises

1. Ca lculate the magnitudes of the following vectors:

(a) v =(2,−1) (b) v =(2,−1,0) (c) v =(3,2,−2) (d) v =(0,0,1) (e) v =(6,4,−4)

2. For the points P =(1,−1,1), Q =(2,−2,2), R =(2,0,1), S =(3, −1,2), does

−−→

PQ =

−−→

RS?

3. For the points P =(0,0,0), Q =(1,3,2), R =(1,0,1), S =(2, 3,4), does

−−→

PQ =

−−→

RS?

4. Le t v =(1, 0,0) and w =(a, 0,0) be vectors in R

. Show that kwk=|a|kvk.

5. Le t v =(a, b, c ) and w =(3a,3b,3c) be vectors in R

. Show that kwk=3kvk.

P(x

, y

, z

)

Q(x

, y

, z

)

R(x

, y

, z

)

S(x

, y

,0)

T(x

, y

,0)

U(x

, y

,0)

Figure 1.1.9

6. Th ough we will see a simple proof of Theorem 1.1

in the next section, it is possible to prove it using

methods similar to those in the proof of Theorem

1.2. Prove the special case of Theorem 1.1 where the

points P =(x

, y

, z

) and Q =(x

, y

, z

) satisfy the fol-

lowing conditions:

> x

>0, y

> y

>0, and z

> z

>0.

(Hint: Think of Case 4 in the proof of Theorem 1.2,

and consider Figure 1.1.9.)

1.2 Vector Algebra 9

1.2 Vector Algebra

Now that we know what vectors are, we can start to perform some of the usual algebraic

operations on them (e.g. addition, subtraction). Before doing that, we will introduce the

notion of a scalar.

Deﬁnition 1.3. A scalar is a quantity that can be represented by a single number.

For our purposes, s calars will always be real numbers.

Examples of scalar quantities are

mass, electric charge, and speed (not velocity).

We can now deﬁne scalar multiplication of

a vector.

Deﬁnition 1.4. For a scalar k and a nonzero vector v, the scalar multiple of v by k,

denoted by kv, is the vector whose magnitude is |k|kvk, points in the same direction as v if

k > 0, points in the opposite direction as v if k < 0, and is the zero vector 0 if k = 0. For the

zero vector 0, we deﬁne k0 =0 for any scalar k.

Two vectors v and w are parallel (denoted by v ∥w) if one is a scalar multiple of the other.

You can think of scalar multiplication of a vector as stretching or shrinking the vector, and

as ﬂipping the vector in the opposite direction if the scalar is a negative number (see Figure

1.2.1).

2v 3v 0.5v

−v

−2v

Figure 1.2.1

Recall that translating a nonzero vector means that the initial point o f the vector is

changed but the magnitude and direction are preserved. We are now ready to deﬁne the

sum of two vectors.

Deﬁnition 1.5. The sum of vectors v and w, denoted by v +w, is obtained by translating

w so that its initial point is at the terminal point of v; the initial point of v +w is the initial

point of v, and its terminal point is the new terminal point of w.

The term scalar was invented by 19

century Irish mathematician, physicist and astronomer William Rowan

Hamilton, to convey the sense of something that could be represented by a point on a scale or graduated ruler.

The word vector comes from Latin, where it means “carrier”.

An alternate deﬁnition of scalars and vectors, used in physics, is that under certain types of coordinate trans-

formations (e.g. rotations), a quantity that is not affected is a scalar, while a quantity that is affected (in a

certain way) is a vector. See MARION for details.

剩余221页未读，继续阅读

patrickpdx

粉丝: 398
资源: 8

Schoolcraft College Michael Corral's Vector Calculus: A One-Sem...

最新资源

Schoolcraft College Michael Corral's Vector Calculus: A One-Sem...

矢量微积分Vector Calculus

vector calculus 6th edition by Jerrold E. Marsden and Anthony Tromba

Brief Vector Calculus.pdf

vector calculus, 6th edition by j.e. marsden and a. j. tromba, published by

vector calculus, linear algebra

vector calculus" by marsden and tromba (6th edition)

vector calculus (6th edition)” by jerrold e. marsden and anthony tromba, fr

calculus pdf

Schema Calculus

the calculus lifesaver pdf

最新资源